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\def\L{\mathord{\b \!\!\! {\rm L}}}
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\def\N{\mathord{\bb\!\!\! {\rm N}}}
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%\advance\count0 by 0
%\pageno=1
\centerline{\huge Mathematical Existence of Crystal Growth}

\centerline{{\huge with Gibbs-Thomson Curvature
Effects}\footnote{}{{\sl 1991 Mathematics Subject Classification.}
Primary 35K05, 49Q20, 80A22
\vskip.03truein}\footnote{}{{\sl Key words and phrases.} crystal growth,
Stefan problem, heat equation, Gibbs-Thomson Law,
geometric measure theory, currents,
varifolds
\vskip.03truein}\footnote{$^{{}^{1}}$}{This work was supported in part by
grants from the National Science Foundation and by
The Geometry Center, University of Minnesota,
a Science and Technology Center funded by the National
Science Foundation, the Department  of Energy, and
and Minnesota Technology, Inc.
\vskip.03truein}} 
\medskip
\centerline{{\larger by Fred Almgren}\footnote{$^2$}{Department
of Mathematics, Princeton University and The National Science
and Technology Research Center for Computation and
Visualization of Geometric Structures
\vskip.03truein}
{\larger and Lihe
Wang}\footnote{$^3$}{Department of Mathematics, Princeton
University and Department of Mathematics, The University of Iowa}}

%\centerline{\large Princeton University}
\bigskip
\bigskip
\centerline{\large ABSTRACT}
\smallskip\noindent
This paper introduces and studies a mathematical evolution
process which models one type of growth of a 
crystal as it freezes from a cold melt.
The crystal freezes (melts) as rapidly as it can anywhere
along its interface where the temperature is below (above)
the local freezing temperature so that rate of growth is
governed by the rate at which latent
heat of fusion can diffuse.  The model incorporates
general Gibbs-Thomson relations between freezing
temperatures and interface surface tension
and general heat capacities and conductivities.
\bigskip \bigskip
\hrule\smallskip\hrule
\bigskip \bigskip
\centerline{\larger Table of Contents}
\bigskip \noindent
{\small
CHAPTER 1. Introduction and Main Theorem.
\medskip \noindent
CHAPTER 2.  Ingredients of our model of crystal growth.

\noindent $\phantom{X}$
2.1 The ambient space.

\noindent $\phantom{X}$
2.2 Crystals.

\noindent $\phantom{X}$
2.3 Heat distributions.

\noindent $\phantom{X}$
2.4 Temperature fields and heat capacities.

\noindent $\phantom{X}$
2.5 Heat conductivity and heat equations.

\noindent $\phantom{X}$
2.6 Surface energy integrands and their integrals.

\noindent $\phantom{X}$
2.7 First variation vector fields for $\Phi$ and $\Phi$ weighted mean curvature.

\noindent $\phantom{X}$
2.8 Gibbs-Thomson relations between curvature and temperature.

\noindent $\phantom{X}$
2.9 Gibbs-Thomson functions and bulk energy densities.

\noindent $\phantom{X}$
2.10 Total energy and minimizers.

\noindent $\phantom{X}$
2.11 Construction of approximating evolutions.

\noindent $\phantom{X}$
2.12 The total energy functions.
\medskip \noindent
CHAPTER 3. Initial conditions.

\noindent $\phantom{X}$
3.1 Proposition.  Some preliminary estimates about our model.

\noindent $\phantom{X}$
3.2 Remarks.

\noindent $\phantom{X}$
3.3 Controls on our evolution process.

\noindent $\phantom{X}$
3.4 Main assumptions about our initial conditions.

\noindent $\phantom{X}$
3.5 Sufficient conditions that our main assumptions about initial conditions
hold.

\noindent $\phantom{X}$
3.6 Solving the heat equation in $\O$ with no crystal present.

\noindent $\phantom{X}$
3.7 Proposition.  An isoperimetric-intersection  estimate.

\noindent $\phantom{X}$
3.8 Proposition.  Isoperimetric inequalities.

\noindent $\phantom{X}$
3.9 Assumptions and terminology.

\noindent $\phantom{X}$
3.10 Proposition.  An estimate relating bulk energy change and Monge-Kantorovich
distance.

\noindent $\phantom{X}$
3.11 Assumptions and terminology.

\noindent $\phantom{X}$
3.12 Corollary.  The evolution when no crystal is present is pure heat flow.

\noindent $\phantom{X}$
3.13 Corollary.  A lower bound on the size of first crystallizations.

\noindent $\phantom{X}$
3.14 Corollary.  A total energy estimate on first crystallizations.

\noindent $\phantom{X}$
3.15 Assumptions and terminology.

\noindent $\phantom{X}$
3.16 Theorem.  A large crystal which appears instantaneously at positive
time.
\bigskip\noindent
CHAPTER 4. Existence and structure of $\EEE$ minimizers.

\noindent $\phantom{X}$
4.1 Theorem.  Existence of $\EEE$ minimizers.

\noindent $\phantom{X}$
4.2 Theorem.  A priori estimates on temperature fields for $\EEE$ minimizers.

\noindent $\phantom{X}$
4.3 Theorem.  Variations of $\EEE$ minimizers.

\noindent $\phantom{X}$
4.4 Theorem.  The Gibbs-Thomson relation for $\EEE$ minimizers when $\Phi$ is 
smooth.

\noindent $\phantom{X}$
4.5 Theorem.  Regularity of $\EEE$ minimizers.
\medskip\noindent
CHAPTER 5.  Heat flows.

\noindent $\phantom{X}$
5.1 Theorem.  Existence of heat flows with fixed crystal.

\noindent $\phantom{X}$
5.2 Theorem.  Inequalities under heat flow.

\noindent $\phantom{X}$
5.3 Theorem.  H\"older estimates for heat flows of approximating evolutions.

\noindent $\phantom{X}$
5.4 Theorem.  Regularity of heat flow across an interface.
\medskip\noindent
CHAPTER 6.  Sufficient conditions for crystal positions and temperature fields to
be close.

\noindent $\phantom{X}$
6.1 Theorem.  Crystal positions, heat distributions, and Dirichlet integrals.

\noindent $\phantom{X}$
6.2 Theorem.  Sufficient conditions for two crystals to be close.
  
\noindent $\phantom{X}$
6.3 Proposition.  Sufficient conditions for two temperature fields to be close.
\medskip\noindent
CHAPTER 7.  Selection of subsequences of approximating evolutions.

\noindent $\phantom{X}$
7.1 Subsequences and Cantor's diagonal process.

\noindent $\phantom{X}$
7.2 Theorem.  General estimates, weakly convergent subsequences, 
and Monge-Kantorovich H\"older continuity of heat distributions.

\noindent $\phantom{X}$
7.3 Terminology.

\noindent $\phantom{X}$
7.4 Theorem.  Sufficient condition that $\l K_\ast(t) \r$,
$Q_\ast(x, \, t)$, $T_\ast(x, \, t)$ be a solution to the heat equation
with varying crystals.

\noindent $\phantom{X}$
7.5 Terminology.

\noindent $\phantom{X}$
7.6 Theorem.  Compactness of the $Y(M)$'s and the modulus of continuity
functions $\sigma_M$.

\noindent $\phantom{X}$
7.7 Main convergence theorem.
\medskip\noindent
CHAPTER 8. The Gibbs-Thomson condition holds for elliptic $\Phi$'s.

\noindent $\phantom{X}$
8.1 Theorem.  Some integral convergences for sequences of crystals.

\noindent $\phantom{X}$
8.2 Theorem.  Some oriented integral estimates over boundaries of crystals.

\noindent $\phantom{X}$
8.3 Theorem.  Bounds on the change in surface integrals of $H(T)$ under heat flow.

\noindent $\phantom{X}$
8.4 Terminology.

\noindent $\phantom{X}$
8.5 Theorem.  An approximate Stokes theorem for the {\rm wmc} integral.

\noindent $\phantom{X}$
8.6 Theorem.  Minimizing properties of the $\l K_\ast(t)\r$'s.

\noindent $\phantom{X}$
8.7 Theorem.  The Gibbs-Thomson condition holds in the limit in the sense
of first variation distributions.

\noindent $\phantom{X}$
8.8 Theorem.  Minimality and regularity of the limit crystals in low dimensions
when $\Phi$ is smooth and elliptic.

\noindent $\phantom{X}$
8.9 Remark.
\medskip\noindent
APPENDIX A.  The bulk energy density function $F$ and the Gibbs-Thomson
function $H$. 

\noindent $\phantom{X}$
A.1 Proposition.  Facts about $F$ and $H$.

\noindent $\phantom{X}$
A.2 Proposition.  Bulk energy functions whose integrals decrease
when heat flows from hot to cold.

\noindent $\phantom{X}$
A.3 Proposition.  The Gibbs-Thomson function effectively determines the
bulk energy function $F$.
\medskip\noindent
APPENDIX B. Measure and function spaces.

\noindent $\phantom{X}$
B.1 A metric space of Radon measures.

\noindent $\phantom{X}$
B.2 The vector space $\L_1(\O)$.

\noindent $\phantom{X}$
B.3 The vector spaces $\L_p(\O)$.

\noindent $\phantom{X}$
B.4 Proposition.  Lower semicontinuity of convex integrals.

\noindent $\phantom{X}$
B.5 Remark.

\noindent $\phantom{X}$
B.6 Corollary.  A compactness estimate.

\noindent $\phantom{X}$
B.7 The vector space $\H^1(\O)$.

\noindent $\phantom{X}$
B.8 Proposition.  Functions in $\H^1(\O)$ have small integrals over
small sets.

\noindent $\phantom{X}$
B.9 Proposition.  A domination estimate.

\noindent $\phantom{X}$
B.10 Proposition.  Sobolev estimates in dimension 2 and 3.
\medskip\noindent
APPENDIX C.  Currents and varifolds and convergence.

\noindent $\phantom{X}$
C.1 Terminology for variations of integrals over currents and varifolds.

\noindent $\phantom{X}$
C.2 Theorem.  Computing first variations.

\noindent $\phantom{X}$
C.3 Theorem.  Current and varifold convergence.

\noindent $\phantom{X}$
C.4 Proposition.  A variant of Jensen's inequality.

\noindent $\phantom{X}$
C.5 Proposition.  A criterion for vector functions to be nearly constant.

\noindent $\phantom{X}$
C.6 Proposition.  Another criterion for vector functions to be nearly constant.
\medskip\noindent
REFERENCES.
}
\bigskip \bigskip
\hrule\smallskip\hrule
\bigskip \bigskip
\centerline{\larger 1. Introduction and Main Theorem}
\bigskip
We introduce and study a mathematical evolution
process which models important and subtle features of
one type of growth of a idealized
crystal as it freezes from a cold melt.

Our main theorem asserts the mathematical
existence of such an evolution in reasonable mathematical spaces.
It is the first such theorem for mathematical models of
crystal growth which allows: (i) general Gibbs-Thomson
relations between interface surface tension and freezing
temperatures, (ii) nonisotropic heat conductivities which
differ in crystal and melt, (iii) heat capacities which differ
in crystal and melt.

When the first author began studying models for crystal growth
it was not clear to him that several
common assumptions about way crystals
may freeze were simultaneously mathematically compatible.
It seemed a stringent condition to impose that crystals and
temperature should interact in such a way that interface
surface tension should exactly
balance temperature.  One
implication of such a requirement is that a temperature maximum
principle cannot hold for heat flow.  Indeed,
on the boundary of a small shrinking crystal with
high curvatures,
temperatures would have to be low and become even lower as the
crystal melts.  Similarly, on the boundary of a small shrinking
hole in a crystal with large curvatures of the opposite sign
temperatures would have to be high and become even higher as
the hole freezes.
The freezing temperature across a planar interface
is what is given in handbooks; we call this temperature
$T_\star$.
Also it was not clear what such a condition should mean when
the evolving crystal was making
complicated geometric or topological changes--dendritic
fingers, for example, can grow into each other.
Finally, equations of heat flow near an evolving crystal
are not standard.
Indeed, the authors still know no general existence theorem
for such equations other than the one proved in this paper.

The mathematical ingredients of the our model are
the following:
\smallskip\noindent$\S$\ 
a torus domain $\O$; since boundary effects of crystal growth
and heat flow are secondary to the main focus of our study it
is easiest to work in a  flat domain without boundary.
\smallskip\noindent$\S$\ 
constant heat capacities for crystal $c_{{}_s}$ and melt $c_{{}_\ell}$
with $c_{{}_s} < c_{{}_\ell}$.  The inequality implies the the
latent heat of fusion depends on temperature and that the
heat equation one must solve is NOT
of the form $\dott = \div\big(\cdots\nabla T \big)$.
\smallskip\noindent$\S$\ 
constant heat conductivities for crystal $\kappa_{{}_s}$ and
melt $\kappa_{{}_\ell}$
Conductivity in the solid is a positive definite
symmetric matrix allowing for the fact that 
the direction of heat flow in some crystalline materials 
need not coincide with the direction of negative temperature gradient.
In the liquid one would typically take the
conductivity matrix to be a multiple of the identity (although
this is not required).
\smallskip\noindent$\S$\ 
a convex surface energy density function $\Phi$ for determining
interface energy and hence interface surface tension.  $\Phi$
is not required to be differentiable.  When $\Phi$ is not
differentiable, weighted mean curvature is not a local
notion, and it is not clear to the authors whether or not the
mathematical implications should be translated into 
Gibbs-Thomson language.
\smallskip\noindent$\S$\ 
a prescribed decreasing Gibbs-Thomson function $H \colon \R^+
\to \R$ which relates interface temperature and interface
surface tension
\smallskip\noindent$\S$\ 
evolving crystals $\l K_\ast(t)\r$;
since we use constructions from geometric measure theory
it is convenient to identify crystals as the integral currents
$\l K \r$ associated with sets of finite perimeter $K$ in
$\O$.
\smallskip\noindent$\S$\ 
evolving temperature fields $T_\ast(x, \, t)$ belonging to
$\L_{2, \, loc}\big(\R^+, \, \H^1(\O)\big)$.
\smallskip\noindent$\S$\ 
evolving heat distributions $Q_\ast(x, \, t)$; since heat
is identified  with internal energy we require that
the integral of $Q_\ast$ over $\O$ should  not change with
time.
\medskip\noindent
With these ingredients we have 
\bigskip
{\large Main Theorem}
{\sl 
Corresponding to a suitable initial crystal $\l K_0 \r$ and
heat distribution $Q$ there exist evolving crystals $\l K_\ast(t) \r$,
evolving heat distributions $Q_\ast(t)$, and associated
evolving temperature fields $T_\ast(t)$  for which the
following is true.
\smallskip\noindent
(1) The heat distributions $Q_\ast(t)$ vary H\"older
continuously with exponent $1/2$ in the Monge-Kantorovich
metric with $Q_\ast(0)= Q_0$ {\rm [this is Conclusion~(2.2) of
Theorem~7.2]}.
\smallskip\noindent
(2) The union of the crystal sets $K_\ast(t)$ is $\LLL$
measurable {\rm [this is Conclusion~(3) of Theorem~7.7]}.
\smallskip\noindent
(3) The crystals $\l K_\ast(t)\r $, heat distributions
$Q_\ast(t)$, and associated temperature fields $T_\ast(t)$
satisfy the heat equations
$$c_{K_\ast} T_\ast = Q_\ast, \quad \dotq_\ast
= \div\big(\kappa_{K_\ast}\, \nabla T_\ast \big)$$
in the sense of distributions {\rm [this is Conclusion~(2) of
Theorem~7.7]}.
\smallskip\noindent
(4) If the surface energy density $\Phi$ is smooth and elliptic,
then, for almost every positive time $t$, the Gibbs-Thomson
condition holds in the sense that
$${d \over ds} \Phi\bigg(G_{s\sharp}\partial \l K_\ast(t) \r
\bigg)\Bigg|_{s=0}
= \int_{x \in K_\ast(t)} \div\left( H\big(T_\ast(x, \, t)\big)
\, g_\ast(x)\right)\, d \LL x $$
for each smooth initial velocity vector field $g_\ast \colon \O \to
\R^n$ and associated deformation $G_s(x) = x + s\, g_\ast(x)$
{\rm [this is Theorem~8.7]}. 
Additionally, if $n=2$ then $\partial K_\ast(t)$ will be a one dimensional
H\"older continuously differentiable submanifold of~$\O$ 
{\rm [this is Conclusion~(2) of Theorem~8.8]}. 
Alternatively, if $n=3$ then $\partial K_\ast(t)$ will be the homeomorphic
image in $\O$ of a compact two dimensional manifold without boundary.
{\rm [this is Conclusion~(3) of Theorem~8.8]}. 
}
\medskip\noindent
Here
\smallskip\noindent$\S$\ 
$\dotq_\ast(t)$ is the time derivative of heat,
\smallskip\noindent$\S$\ 
$\nabla T_\ast(t)$ is the space gradient of temperature,
\smallskip\noindent$\S$\ 
$c_{K_\ast(t)}$ is the heat capacity function in $\O$ equal
to $c_{{}_s}$ in $K_\ast(t)$ and to 
$c_{{}_\ell}$ in $\O \sim K_\ast(t)$,
\smallskip\noindent$\S$\ 
$\kappa_{K_\ast(t)}$ is the heat conductivity function equal
to $\kappa{{}_s}$ in $K_\ast(t)$ and to 
$\kappa_{{}_\ell}$ in $\O \sim K_\ast(t)$,
\smallskip\noindent
In case $H(T_\ast(t))$ is $\HH\rightangle \partial
K_\ast(t)$ summable (which is not asserted in general),
then the Gauss-Green  theorem tells us
$${d \over ds} \Phi\bigg(G_{s\sharp}\partial \l K_\ast(t) \r
\bigg)\Bigg|_{s=0}
= \int_{x \in \partial K_\ast(t)} \bigg(H\big(T_\ast(x, \, t)\big)
\, {\bf n}_{K_\ast(t)}(x)\bigg) \bullet g_\ast(x)\, d \LL x $$
in conclusion (4) above; here
${\bf n}_{K_\ast(t)}(x)$ is the unit exterior normal  vector
to $K_\ast(t)$ at~$x$.
This equation means that
$H\big(T_\ast(x, \, t)\big) \, {\bf n}_{K_\ast(t)}(x)$
is the $\Phi$ weighted mean curvature of $\partial K(t)$ in
the sense of first variation distributions [AW].
\bigskip
We give an example in Theorem~3.16 of initial conditions which lead
to a discontinuity in crystal positions at positive time.
\bigskip
The evolution process of the theorem
is the limit as $j \to \infty$ of approximating
evolutions $\l K_j \r$, $T_j$, $Q_j$
associated with time steps $\Delta t_j$ converging to zero. 
This approximating evolutions are obtained by alternating 
a heat flow with fixed crystal for a time $\Delta t_j$ together
with a crystal changing step which occurs with the clock stopped.

{\bf The heat flow process.} During each time interval
the crystal $K = K_j$ remains constant while
temperatures fields $T_j$ and heat distributions
$Q_j$ evolve according to the heat equations
$$c_{K} T_j = Q_j, \quad \dotq_j
= \div\big(\kappa_{K}\, \nabla T_j \big).$$
The existence of such heat flow is established in Theorem~5.1.  In
Theorem~5.2 we establish a regularity theorem for such heat flow
in case the crystal boundary is a H\"older continuously differentiable 
hypersurface.

{\bf The crystal moving step.} At the end of a $\Delta t_j$
time interval, one chooses new a crystal position $\l L \r$
and adjusted heat distribution $Q$ by using general existence
theorems from geometric measure theory to minimize the quantity
$$\Phi\big(\partial \l L \r\big)
+ \int_\O c_L F(T)\, d \LL + {1 \over \Delta t^\alpha} \| P - Q
\|_\ast.$$
Existence and properties of the new crystals and temperature fields
resulting from the crystal moving step are set forth in Chapter~4.  In 
Theorem~4.5 we show in dimensions~2 and~3, that crystal interfaces
are H\"older continuously differentiable hypersurfaces in case the
surface energy function~$\Phi$ is smooth and elliptic.
Here
\smallskip\noindent$\S$\ 
$\Phi\big(\partial \l L \r\big)$ is the surface energy of the
liquid solid interface.
\smallskip\noindent$\S$\ 
$\int_\O c_L F(T)\, d \LL$ is the bulk energy associated with
the crystal $\l L\r$ and the heat distribution $Q$.
$L$ together with $Q$ determine the new temperature $T$.  The
bulk energy function $F \colon \R^+ \to \R^+$ is determined
by the desired Gibbs-Thomson function $H$ as above.
$F$~is convex and has a minimum at the planar freezing
temperature~$T_\star$.
\smallskip\noindent$\S$\ 
$\alpha$ is a small positive number less than $1/24$.
This bound on the size of $\alpha$ is determined by the requirement
that the exponent $1/12 - 2 \alpha$ of $\Delta t$ in the conclusion
inequality of Theorem~8.3 be a positive number.
\smallskip\noindent$\S$\ 
$P$ is the heat distribution which was present at the end of the
heat evolution of the previous time interval.
\smallskip\noindent$\S$\ 
$\|P-Q\|_\ast$ is the Monge-Kantorovich distance between the
heat  distributions $P$ and $Q$; it equals the infimum of the
numbers
$$\int_\O \big(Q - P) \varphi\, d \LL$$
over all test functions $\varphi$ defined on $\O$ which
satisfy a Lipschitz condition with constant not  exceeding
one.  This is the first time to the
knowledge of the authors that a Monge-Kantorovich
distance between heat distributions has been used in parabolic
problems.  Its introduction makes possible major
simplifications in the technical details of our arguments.

It is necessary to have some way of moving heat before
computing bulk energy in order for this  variational scheme
to work.  Consider the case of water
freezing.  The bulk energy function $F(T)$
has its minimum at $T_\star$ which for water corresponds to
$0^\circ$C.  Suppose a water-ice interface were at say
temperature $T_1$ corresponding to $-0.1^\circ$C.  If one could
freeze a small region of water into ice and use the
latent heat of fusion to warm nearly water or ice
to nearly $0^\circ$C then the bulk energy integral would decrease.
This is the main driving mechanism for freezing in our model.
If the same heat (internal energy) remained in the region just
frozen, the heat of fusion of 80 calories per gram would raise
the temperature of the newly frozen ice to a much higher
temperature $T_2$ with $F(T_2)>>F(T_1)$
so that there would be a large bulk energy disadvantage in freezing.
Hence, there is a need to find a reasonable
way of smearing out the heat  of
fusion before computing bulk energies.
It must be done enough so that crystals do
grow reasonably, but not so much that the cumulative effects of
the smearing out means that the heat equations are not
satisfied in the limit--each smearing out itself is a
discontinuity in the heat flow process.  Monge-Kantorovich
methods seem well adapted to such control.  With the
Monge-Kantorovich term present in the  quantity being
minimized, it turns out that the Gibbs-Thomson relation holds
exactly after each crystal moving.  

There is an appealing conceptual generality as well as various
practical advantages in the use of
variational schemes to model crystal growth with Gibbs-Thomson
effects.
By formulating the problem in the geometric measure
theory setting for static minimizations one can invoke
existence and structure theorems of considerable power and
generality and easily work with the complicated geometric
configurations such as arise in  dendritic structures.
Also in  this setting one does not have to make specific
reference to principal or mean curvatures.  This is of
theoretical importance because there will be times when these
are not classically defined.  It is of practical importance
in computations because calculation of curvature is
notoriously difficult and  unreliable.

Such a variational scheme was first suggested to the first author
several years ago by R. Almgren when he 
discovered a conceptual flaw in one of the
algorithms the first author had been considering to compute
crystal growth.  This led the first author 
within several months to an existence theorem for growth
with general surface energies provided heat conductivities and
capacities were uniform.  Independently, and
several months earlier, S. Luckhaus [LS] discovered a similar
construction and obtained an existence theorem in case
surface energy equaled surface area. 
These schemes
together with that of the present paper are the only ones
known to the authors which successfully model crystal growth
with Gibbs-Thomson curvature effects.
The assumption of uniform heat capacities
substantially limited the applicability of the earlier results; 
the heat capacities of ice and water, for
example, differ by a factor of two.    

Some of the mathematical assumptions (like
constant heat capacities and heat conductivities in both solid
and liquid) we make in
our model have physical relevance only in a limited range of
temperatures about the usual freezing temperature 
$T_\star$ and are perhaps best thought of as
a ``linearization'' of more complicated,  more nonlinear
relationships whose exact nature is generally unknown.

The pattern into which our crystal grows is often the result
of a competition
between a (generally destabilizing) 
bulk energy pressure to freeze cool liquid (or melt warm
solid) and an (generally stabilizing)
surface tension to reduce interface energy.
Suppose, for example, a crystal were growing into cold melt
with nearly uniform velocity along a nearly planar interface and a
bump developed along the interface sticking into the
cold melt.  This bump would have a relative
advantage diffusing away heat of fusion; it would also
have increased surface tension with associated
slightly depressed freezing temperature.
If the diffusion
advantage prevails then growth along the bump would accelerate
perhaps creating a dendrite; this is one form of the
``Mullens-Sekerka instability''.  If surface tension
prevailed the bump would shrink.
R. Almgren has written code to implement his variational
algorithm for ``crystal growth'' in the plane.  His
computations and figures nicely
illustrate the dependence of the dendritic patterns of
growth on the conditions of growth [AR].  
Note also the computations of A. Roosen [RA].

In our model, the plot of heat $Q$ as a function of
temperature $T$ in the
solid (resp. liquid) is a straight line having slope $c_{{}_s}$
(resp. $c_{{}_\ell}$) since heat capacity is the 
amount of heat required to raise temperature one unit.
Also, in our model, at the freezing temperature $T_\star$.
heat energy (internal energy) is greater in the liquid than
in the solid and heat energy is liberated
when liquid solidifies at this temperature.
This means that our two straight line graphs
intersect at some temperature less
than $T_\star$.  We choose our temperature scale so that this
temperature is zero by definition.  This is substantial 
mathematical convenience and is not supposed to be associated
with absolute zero in the physical world.  We do construct our
bulk energy function $F$ so that temperatures lower than zero
will not occur.

Among the general mathematical models of crystal growth
discussed in the overview [CHT] by J. Cahn,
C. Handwerker, and J. Taylor is the formula
$$v = M(H + \wmc)$$
in which $v$ denotes the surface normal velocity, $H$ is
(some reasonable function of the)
undercooling below the freezing temperature $T_\star$,
$\wmc$ is the weighted mean curvature
or surface tension of the interface,
and $M$ is the mobility of the interface 
measuring the response of the interface to the driving pressures
of undercooling and surface tension, e.g. it make take time
for a cold liquid to assemble its atoms into a crystal
lattice.  This equation also applies to a
wide range of other interface motion problems, involving chemical- as
well as temperature-dependent bulk driving forces.
There seems little general theoretical or experimental knowledge
about the form of the mobility function $M$ (only its sign
seems determined by classical thermodynamics [GM]).
If one takes $H$ identically equal to zero and $M$
identically equal to one, then one obtains a heuristic 
equation for ``motion by weighted mean curvature''; a
variational scheme similar to the present one was used by the
authors together with J.~Taylor in the study of such motion [ATW].
In the present paper the
mobility $M$ is infinite; this is interpreted to 
mean that bulk and interface forces must be in equilibrium
across the  interface.  It is in this sense that we say that
freezing and melting occur instantaneously across along
the interface.  In calculations of crystal growth by various
schemes one {\sl de facto} has finite mobility.  
The present existence theorem suggests that
calculations need not become wildly unstable
as mobility becomes large.

During the preparation of this paper the
first author was a visitor at the
Centre for Mathematical Analysis of the
The Australian National University and at the
Aspen Center for Physics and was a 
visiting member of the Institute for Advanced Study,
Princeton.
\bigskip
\centerline{\larger 2. Ingredients of our model of crystal growth}
\bigskip
In  this chapter we give formal definitions to establish the
mathematical setting in which we study crystal growth.  We
discuss first variations of surface energy
and Gibbs-Thomson relations and set forth the time stepping
procedure for constructing approximating evolutions.
\bigskip
{\bf 2.1 The ambient space.}  Corresponding to a fixed (large) positive
number $\rho_\ast$ we denote by $\O$ the Cartesian product (as Riemannian manifolds)
of $n$~circles, each of circumference $\rho_\ast$.  By identification of
$\O$ with the quotient of $n$~dimensional Euclidean space $\RR$ by $\rho_\ast$ 
times the integer lattice we obtain local coordinate functions $x_1, \, \ldots, \, x_n$, 
an inner product $v \bullet w$ between various tangent vectors $v$ and $w$, 
orthonormal basis vectors ${\bf e}_1, \, \ldots, \, {\bf e}_n$  for tangent spaces,
and a distance $|p - q|$ between points $p$ and $q$ in $\O$.
The diameter of $\O$ is 
$$\hbox{diam}(\O) = {1\over 2} n^{1 \over 2} \rho_\ast.$$
In the obvious way, we 
measure $\O$ by Lebesgue's $n$~dimensional measure $\LL$ and by Hausdorff's
\hbox{$n-1$}~dimensional measure $\HH$ so that the volume of $\O$ is 
$$|\O| := \LL(\O) = \rho_\ast^n.$$ 

One reason that we work in $\O$ rather than in $\RR$ is to be able to treat 
questions of heat flow without the complications
of boundary effects (equivalently, we are using periodic boundary conditions).
Another reason is to avoid questions about the way in which crystallization should
occur at the boundary.

When we are studying time evolutions we will use the space-time $$\O\times \R^+$$ with 
coordinate functions $x_1, \, \ldots, \, x_n, \, t$.
For functions $T$ defined on $\O$ or on $\O\times \R^+$ the gradient notation $\nabla T$
always refers to (space) gradients computed with respect to coordinates 
$x_1, \, \ldots, \, x_n$ only (not $t$).  For functions $Q$ defined on 
$\O\times \R^+$ the dot notation $\dotq$ always refers to $t$ (time) derivatives.

We denote by ${\cal D}\big(\O\big)$ the real vector space of
all infinitely differentiable
functions $\varphi \colon \O \to \R$.  Similarly, for
$-\infty \le a < b \le \infty$ we denote by ${\cal D}\big(\O \times
(a, \, b)\big)$ the real vector space of all infinitely differentiable
functions $\varphi \colon \O \times (a, \, b) \to \R$ having compact
support.
\bigskip
{\bf 2.2 Crystals.}  The crystals we consider are all integral currents
$\l K \r$ associated with $\LL$ measurable subsets $K$ of $\O$ having finite 
perimeter.  Currents seem the natural setting for our study since, in particular,
sets differing in measure zero are identified.  Associated
with each such $K$ is its perimeter $\partial K$ consisting those points
$p$ at which $K$ has a (measure theoretic) unit exterior normal vector ${\bf n}_{{}K}(p)$.
As currents we can write 
$$\partial \l K \r = \HH \rightangle \partial K \wedge *{\bf n}_{{}K}.$$
We fix positive a number $0 < \gamma_\star < 1$ 
and denote by
$$\KK$$
the collection of all $\l K \r$'s for which the following two
conditions hold:
\smallskip\noindent
{\bf 2.2.1}  One or the other inequalities 
$$\M(\l K \r ) = \LL(K) \le \gamma_\star |\O|\quad \hbox{\bf
OR}\quad
\M(\l K \r ) = \LL(K) \ge \gamma_\star |\O|$$  
holds for all members of $\KK$.  See the discussion of initial
conditions in Chapter~3.  As noted there
this assumption is used in Step~1 of Part~3 of
the proof of Theorem~6.1 which theorem provides a
basic estimate in showing convergence of our
approximating evolutions given in 2.11 below.
\smallskip\noindent
{\bf 2.2.2} $\M(\partial K) = \HH(\partial K) \le B_\star$; here
as a convenient choice we set
$$B_\star = \EE_0/\inf\big\{ \Phi(v)\colon |v| = 1\big\}$$
as determined by our surface energy function $\Phi$ in~2.6
below and by our initial total energy $\EE_0$ in~2.12 below.

\noindent
By a {\bf crystal} in $\O$ we mean any member of $\KK$.  If $\l K \r$ is a crystal 
then the current $\l \O \sim K \r$ is called the {\bf melt of} $K$.

The metric we use on $\KK$ is given by the mass norm (on currents) so that the 
distance between $\l K \r$ and $\l L \r$ in $\KK$ equals
$$\M(\l K \r - \l L \r) = 
\LL\big( K \tt L \big).$$
Here 
$$K \tt L =  (K \sim L ) \cup ( L \sim K)$$
is the symmetric difference between $K$ and $L$.
The space $\KK$ is compact in the mass topology.

\bigskip
{\bf 2.3 Heat distributions.}  By a {\bf heat (internal energy) 
distribution}
in $\O$ we mean any (equivalence class of) nonnegative
function(s) $Q$ in $\L_2(\O)$ for which $1/Q$ also belongs to $\L_2(\O)$ and for which
$$\int_\O Q \, d \LL  = Q_\star  |\O|.$$
Here $Q_\star$ is a fixed positive number specified in giving initial conditions
in Section~3.2 below.
We denote by 
$$\QQ$$
the affine subspace of $\L_2(\O)$ of heat distributions.  One important metric
on $\QQ$ associates with heat distributions $P$ and $Q$ in $\QQ$ the
{\bf Monge-Kantorovich distance}
$$\| P - Q \|_\ast = \sup \left\{ \int_\O \varphi \big( P - Q \big) \, d \LL
\colon \lip (\varphi) \le 1 \right\}.$$
Here
$$\lip (\varphi) = \sup\left\{ {|\varphi(x) - \varphi(y)|\over |x - y|}\colon
x, \, y \in \O \hbox{ with } x \not= y \right\}$$
is the Lipschitz constant of $\varphi$.  The review article
[RS] by S. T. Rachev gives relations between our definition above
and the ``Monge optimization problem'' and ``Kantorovich's mass
transference problem''.

For any pair $P$ and $Q$ of heat distributions in $\QQ$ we
have
$$\| P - Q \|_\ast \le \hbox{diam}(\O)\,Q_\star|\O| =
2^{-1}n^{1\over2}\rho_\ast Q_\star|\O|.$$

For each positive number $M$ the set
$$\QQ \cap \big\{ Q \colon \|Q \|_{{}_{{\bf L}_2}} + 
\| 1/Q \|_{{}_{{\bf L}_2}} \le M\big\}$$
is compact in the $\| \cdot \|_\ast$ topology as noted in Appendix~B.6. 
The use of this distance function in a parabolic problem is a novel 
feature of the present paper.  It is also indispensable in our proofs.

\bigskip
{\bf 2.4 Temperature fields and heat capacities.}  By a {\bf temperature field}
in $\O$ we mean any (equivalence class of) nonnegative function(s) in $\L_2(\O)$.
It is an assumption in our model that there is a linear relationship between
heat and temperature in both crystal determined by positive coefficients
$0 < c_{{}s} < c_{{}\ell}< \infty$.
Corresponding to each $\LL$ measurable subset $K$ of $\O$ (in
particular to each crystal $\l K \r$ in $\KK$) we define
$$c_K \colon \O \to \{ c_{{}_s} , \, c_{{}_\ell}\}, \quad
c_{{}K}(x) = \cases{c_{{}s} & if $x \in K$ \cr
                    \phantom{X} & \cr
                    c_{{}\ell} & if $x \in \O \sim K$\cr}.$$
In our model we always assume that the relationship 
$$c_{{}K}T = Q$$
holds $\LL$ almost everywhere between temperature $T$, heat $Q$, and crystal $\l K \r$.
The number $c_{{}s}$ is thus the {\bf heat capacity of the solid crystal}
while $c_{{}\ell}$ is the {\bf heat capacity of the liquid melt}.

If $c_{{}K}T = Q \in \QQ$ as above with $T \in \H^1(\O)$ then
Poincar\'{e}'s inequality as in appendix B.7 guarantees
$$\int_\O T^2 \le 
{Q_\star^2|\O|^2 \over \rho_\ast^n c_s^2}
+ \rho_\ast^2 \int |\nabla T|^2 \, d \LL .  $$
\bigskip
{\bf 2.5 Heat conductivity and heat equations.}  We assume that the
flow of heat (a ``velocity vector field'') in response to a temperature
gradient (a vector field) is determined by positive definite symmetric conductivity matrices
$\kappa_{{}_s}$ and $\kappa_{{}_\ell}$.  
We set
$$\kappa_0 = \sup \big\{ \| \kappa_{{}_s}\|, \, \| \kappa_{{}_\ell}\| \big\},$$
(mapping norms)
and denote by $ \kappa_\ast$
the smaller of the two numbers
$$\inf \big\{v \bullet  \kappa_{{}_s}(v) \colon v \in \RR  \hbox{ with }|v| = 1\big\}, \quad
\inf \big\{v \bullet  \kappa_{{}_\ell}(v) \colon v \in \RR  \hbox{ with }|v| = 1\big\}$$
and denote by $\kappa^\ast$
the larger of the two numbers
$$ \sup \big\{v \bullet  \kappa_{{}_s}(v) \colon v \in \RR \hbox{ with }|v| = 1 \big\}, \quad
\sup \big\{v \bullet  \kappa_{{}_\ell}(v) \colon v \in \RR \hbox{ with }|v| = 1 \big\}.$$
Corresponding to each crystal $\l K \r$ 
in $\KK$ we define 
$$\kappa_{{}_K} \colon \O \to \left\{ c_{{}_s}, \, c_{{}_\ell} \right\}, \quad
\kappa_{{}_K}(x) =\cases{\kappa_{{}_s} & if $x \in K$\cr
                           \phantom{X} & \cr
                           \kappa_{{}_\ell} & if $x \in \O \sim K$.\cr}$$
We say that functions $Q(x, \, t)$ and $T(x, \, t)$
defined on $\O \times [a, \, b]$ are a {\bf solution to the heat equation
with fixed crystal $\l K\r$} and with initial heat distribution $Q_a$ and final heat
distribution $Q_b$ if and only if the equations
$$c_{{}_K} T = Q, \quad \dotq = \div\left( \kappa_{{}K}\nabla T\right),
 \quad  Q(\cdot, \, a) = Q_a, \quad Q(\cdot, \, b) = Q_b$$
hold in the following sense:
\smallskip\noindent
{\bf 2.5.1}  $\l K \r$ is a crystal in $\KK$.
\smallskip\noindent
{\bf 2.5.2}  The functions $Q_a$ and $Q_b$ and $Q(\cdot, \, t)$ belong to $\QQ$
for each $a < t < b$.
\smallskip\noindent
{\bf 2.5.3}  The function $Q$ belongs to $\C^0\big[(a, \, b), \, \L_2(\O)\big]$.
\smallskip\noindent
{\bf 2.5.4}  The function $T$ belongs to $\L_2\big[(a, \, b), \, \H^1(\O)\big]$;
this implies the existence of $\nabla T$ in $\L_2\big[ (a, \, b) , \, \L_2(\O, \, \RR)\big]$.

\noindent
For each $\varphi \in {\cal D}\big(\O \times (a, \, b)\big)$
the following two equalities hold
$$0 = \int_{\O \times (a, \, b)} \varphi \nabla T  + \nabla \varphi T \, d \LLL
\leqno{{\bf 2.5.5}}$$
(this is actually implied by 2.5.4).
$$0 = \int_{\O \times (a, \, b)} \varphi \big( c_{{}_K}T - Q\big)
 \, d \LLL\leqno{{\bf 2.5.6}}$$

\noindent
For each $\varphi \in {\cal D}\big(\O \times \R \big)$
the following equality holds
$$0 = \int_{\O \times (a, \, b)} -\dotf Q + \kappa_{{}_K}(\nabla \varphi)\bullet
\nabla T\, d \LLL + \int_\O \varphi(\cdot, \, b) Q_b - \varphi(\cdot, \, a) Q_a \, d \LL.
\leqno{{\bf 2.5.7}}$$
The existence, properties, and regularity of
such solutions are discussed in Chapter~5.

Similarly, we say that crystals $\l K(t) \r$ and functions $Q(x, \, t)$, and $T(x, \, t)$
defined for $a\le t \le b$  are a 
{\bf solution to the heat equation with varying crystals}
with initial heat distribution $Q_a$ and final heat
distribution $Q_b$ if and only if the equations
$$c_{{}_K} T = Q, \quad \dotq = \div\left( \kappa_{{}_K}\nabla T\right), \quad
 Q(\cdot, \, a) = Q_a, \quad Q(\cdot, \, b) = Q_b$$
hold in the following sense:
\smallskip\noindent
{\bf 2.5.8} Each $\l K(t)\r$ is a crystal and the set
$\big\{ (x, \, t) \colon a < t < b, x \in K(t) \big\}$ is $\LLL$ measurable.
\smallskip\noindent
{\bf 2.5.9}  The function $Q$ belongs to $\L_{2, loc}\big[(a, \, b), \, \L_2(\O)\big]$.
\smallskip\noindent
{\bf 2.5.10}  The functions $Q_a$ and $Q_b$ and $Q(\cdot, \, t)$ belong to $\QQ$
for each $a \le t \le b$, and the function $[a, \, b] \to \QQ$ which maps
$t$ to $Q(\cdot, \, t)$ is continuous in the Monge-Kantorovich distance
topology with 
$\| Q_a - Q(\cdot, \, a) \|_\ast = 0$
and
$\| Q_b - Q(\cdot, \, b) \|_\ast = 0$


\smallskip\noindent
{\bf 2.5.11}  The function $T$ belongs to $\L_2\big[(a, \, b), \, \H^1(\O)\big]$;
this implies the existence of $\nabla T$ in $\L_2\big[ (a, \, b) , \, \L_2(\O, \, \RR\big)]$.

\noindent
For each $\varphi \in {\cal D}\big(\O \times (a, \, b)\big)$
the following two equalities hold:
$$0 = \int_{\O \times (a, \, b)} \varphi \nabla T  + \nabla \varphi T \, d \LLL
\leqno{{\bf 2.5.12}}$$
(this is actually implied by 2.5.11).
$$0 = \int_{(x, \, t) \in \O \times (a, \, b)} \varphi(x, \, t)
\big( c_{{}_{K(t)}}(x) \,T(x, t) - Q(x, \, t) \big) \, d \LL x d {\cal L}^1 t)
\leqno{{\bf 2.5.13}}$$
For each $\varphi \in {\cal D}\big(\O \times \R\big)$
$$\eqalign{0 = \int_{(x, \, t) \in \O \times (a, \, b)} &
 - \dotf(x, \, t) Q(x, \, t) \cr
&+ \kappa_{{}_{K(t)}}(x) \left(\nabla \varphi(x, \, t)\right)\bullet
\nabla T(x, \, t) \, d \LL x\, d {\cal L}^1 t \cr
&\phantom{XXXXX} + \int_{x \in \O} \varphi(x, \, b)\, Q_b(x) - \varphi(x, \, a)\, Q_a(x)
 \, d \LL x .\cr} \leqno{{\bf 2.5.14}}$$

The matrix $\kappa_{{}_s}$ is thus the {\bf heat conductivity of the solid crystal}
while $\kappa_{{}_s}$ is the {\bf heat conductivity of the liquid melt};  since
$\O \sim K$ is a liquid one usually assumes 
$\kappa_{{}_\ell}$ is a positive multiple of the $n \times n$~identity matrix.

\bigskip
{\bf 2.6 Surface energy integrands and their integrals.}
We denote by
$$\Phi \colon \RR \to \R^+$$
a fixed {\bf parametric surface energy integrand.}  This means that $\Phi$
is continuous and positively homogeneous of degree one (i.e. $\Phi (\lambda v) = \lambda
\Phi (v)$ for each $v$ and each nonnegative $\Phi$).  We also assume that
$\Phi$ is positive on $\RR$ except at the origin and that $\Phi$ is 
convex.\footnote{$^1$}{This assumption of convexity is not necessarily true for
the surface energy functions of physical crystalline materials.  When energies
are not convex it is customary to replace the actual energy by the smallest convex
energy which is not less that the actual energy and then understand surfaces
to have tangent planes in the sense of varifold geometry.  The Wulff shape of
the convexified energy is the same as that of the original energy.}  We call
$\Phi$ {\bf smooth} provided the function $\Phi$ is three times H\"older
continuously differentiable except at the origin.  We call a smooth $\Phi$
{\bf elliptic} provided $\Phi$ is uniformly convex in the sense that the 
restriction of $\Phi$ to any arc length parametrized line in $\RR$ not containing
the origin has strictly positive second derivatives.

Whenever $\l K \r$ is a crystal in $\KK$ we call the number
$$\Phi\big(\partial \l K \r \big):= \int_{x \in \partial K}\Phi\left( {\bf n}_K(x)
\right)\, d \HH x$$
the {\bf integral of $\Phi$ over $\partial \l K \r$}.
\bigskip
{\bf 2.7  First variation vector fields for $\Phi$ and $\Phi$ weighted mean curvature.}
Associated with each smooth initial velocity 
vector field $g \colon \O \to \RR$ is the one parameter of diffeomorphisms
$$G_t\colon \O \to \O, \quad G_t(x) = x + t g(x)$$
defined for small times $t$.  Associated with each crystal
$\l K \r$ in $\KK$ and each such initial velocity vector field $g$ is the 
{\bf first variation}
$${d \over dt} \Phi\left( G_{t\sharp}\partial \l K \r \right) \bigg|_{t=0}$$
which always exists in case $\Phi$ is smooth.  It may occur (and will often
be the case in this paper) that there is a vector field
${\bf H} \colon \partial K \to \RR$
$${d \over dt} \Phi\left( G_{t\sharp}\partial \l K \r \right) \bigg|_{t=0}
= \int_{ x \in \partial K}  {\bf H}(x) \bullet  g(x)\, d \HH x$$
for every initial velocity vector field $g$.  In that case, {\bf H} is
defined $\HH$ almost uniquely, and one says that {\bf H} is the
{\bf $\Phi$~first variation vector field of $\partial \l K \r$}.  When one
can write ${\bf H} = wmc \, {\bf n}_{{}_K}$, the function $wmc$ is called
the {\bf $\Phi$~weighted mean curvature of $\partial \l K \r$}.\footnote{$^2$}{In
case $\Phi(v) = |v|$ surface energy coincides with surface area.  In that
case the weighted mean curvature of the unit sphere in $\R^3$ bounding
the unit ball as crystal has weighted mean curvature equal to two everywhere.
The mean curvature vector field in the sense of differential geometry consists
of inward point normal vectors of unit length.}
\bigskip
{\bf 2.8 Gibbs-Thomson relations between curvature and temperature.}
When we say that a crystal $\l K \r$ in $\KK$ and a temperature field
$T$ satisfy the {\bf Gibbs-Thomson relation} with
Gibbs-Thomson function $H$ we mean that the following is true.
\smallskip\noindent
{\bf 2.8.1} $H \colon \R^+ \to \R$ is a smooth decreasing function, 
\smallskip\noindent
{\bf 2.8.2} The vector field $H\big(T(\cdot)\big)\, {\bf n}_{{}_K}(\cdot) \colon
\partial K \to \RR$ is the $\Phi$~first variation vector field of $\partial \l K \r$.
In other words, a Gibbs-Thomson relation holds if and only if weighted
mean curvature equals a prescribed function of temperature.
\bigskip
{\bf 2.9 Gibbs-Thomson functions and bulk energy densities.}  The object of
this paper is to use a variational scheme of minimizations of sums of bulk 
energies and surface energies in order to obtain
solutions to the heat equation with varying crystals for which
the Gibbs-Thomson relation between weighted mean curvature 
and temperature holds at almost every time (at least
when $\Phi$ is smooth).  We seek to do this for any suitable Gibbs-Thomson
function $H$.  The nonnegative 
{\bf bulk energy function} $F(T)$ used in the minimizations is determined by
the Gibbs-Thomson function $H$ 
as a solution to the ordinary differential equation
$$H(T) = \left( c_{{}_\ell} - c_{{}_s}\right) \bigg( F(T) - T F'(T)\bigg)\leqno{{\bf 2.9.1}}$$
with initial condition
$$F(T_\star) = 0\leqno{{\bf 2.9.2}}$$
for a positive temperature $T_\star$.
In our model
the {\bf bulk free energy density of liquid} at temperature $T$
 is $c_{{}_\ell} F(T)$ while
the {\bf bulk free energy density of solid} is $c_{{}_\ell} F(T)$.
In the present setup the bulk free
 energy densities have a unique minimum at temperature
$T_\star$ and if the crystal has an interface near a point which is planar, then
our sum of bulk and surface energies cannot be decreased by local changes in
crystal position.  For this reason we call
$T_\star$ the {\bf standard freezing temperature} of our crystalline substance
Such an equilibrium condition corresponds to the physical definition of
standard freezing temperature.

Figure~1 illustrates the relations between heat (internal)
energy density and temperature assumed in this model.  The
slope of line QL is $c_{{}_\ell}$ and the slope of line
QS is $c_{{}_s}$.  The standard freezing temperature $T_\star$ is
denoted T0 in Figures 1 and~2.  The latent heat of fusion at
the standard freezing temperature $T_\star$
is thus $\big( c_{{}_\ell} -c_{{}_s}\big) T_\star$.


Figure~2 illustrates bulk free energies.  The curve EL is the graph
of the function
$c_{{}_\ell}F(T)$
while the curve ES is the graph of the function
$c_{{}_s}F(T)$.

There are several conditions which we impose on $F$ (correspondingly
on $H$) for
the purposes of our mathematical analysis as follows:
\smallskip\noindent
{\bf 2.9.3} $F$ is twice continuously differentiable and
there is a positive number $c_\ast$ such that
$$F''(T) \ge  2 c_\ast$$
for each $T$, so that $F$ is uniformly convex.
\smallskip\noindent
{\bf 2.9.4}  There are numbers $0 < a_\ast < T_\star < b_\ast < \infty$ such that
$$F(T) = \cases{ T^{-2} & if $0 < T \le a_\ast$\cr
\phantom{X} & \cr
T^2 & if $ b_\ast \le T < \infty$.\cr}$$
These conditions on the behavior of $F$ at extremes of temperature are primarily
for mathematical convenience.  Indeed, such temperatures  are beyond the range at which
one would normally wish to think of the model as a representative of physical
reality.  In order to have a mathematical theory, however, we have to do something; 
we might as well do something convenient.  We actually have very few restrictions
on the possible behavior of $H$'s as is asserted by the following fact which is a
consequence of a standard existence theorem for ordinary differential equations
(see Appendix~A.3) and a straight forward construction.
\smallskip\noindent
{\bf 2.9.5 Fact.}  Suppose $H_0 \colon \R^+ \to \R$ is any continuously differentiable
decreasing function such that $H_0'(T)$ does not vanish and such that $H_0(T_\star) = 0$.
Then, corresponding to each $0 < a < T_\star < b < \infty$, there exist $0 < a_\ast < a < T_\star
< b < b_\ast < \infty$ and $c_\ast > 0$ and functions $H$ and $F$ as above
(satisfying conditions 2.9.1, 2.9.2, 2.9.3, 2.9.4) such that
$H(T) = H_0(T)$ for each $a \le T \le b$.  
\smallskip\noindent
We henceforth assume that functions $H$ and $F$ satisfying conditions
2.9.1, 2.9.2, 2.9.3, 2.9.4 above are fixed.  We also fix the positive
constant $C_\ast$ given in Appendix~A.
\bigskip
{\bf 2.10 Total energy and minimizers.}  
Associated with each crystal $\l K \r$ in $\KK$
and each heat distribution $Q$ is the {\bf total energy }
$$\EEE\big(\l K \r, \, Q )
 = \Phi \big( \partial \l K \r \big) + 
\int_\O c_{{}_K}F\left( c_{{}_K}^{-1}Q \right) \, d \LL
 = \Phi \big( \partial \l K \r \big) + 
\int_\O c_{{}_K}F\left( T \right) \, d \LL$$
which is a sum of surface energy and bulk energy;
here $T= c_{{}_K}^{-1}Q$ is the temperature field associated with the crystal
$\l K \r$ and the heat distribution $Q$.

When we say that $\l K \r$, $Q$ {\bf give an} $\EEE$ {\bf minimizer for}
$P$ {\bf and} $\Delta t$
we mean that $\l K \r$ is a crystal in $\KK$, $P$ and $Q$ are heat distributions
in $\QQ$, $\Delta t$ is a positive time interval, and
$$\EEE\big(\l K \r, \, Q ) + { 1 \over \Delta t^\alpha} \| Q - P \|_\ast \le
\EEE\big(\l L \r, \, R ) + { 1 \over \Delta t^\alpha} \| R - P \|_\ast$$
whenever $\l L \r$ is a crystal in $\KK$ and $R$ is a heat distribution in $\QQ$.
Here $\alpha$ is any fixed small positive number less than $1/24$. 
\bigskip
{\bf 2.11 Construction of approximating evolutions.}  Suppose that $\l K_0\r$ is
a initial crystal in $\KK$ and $Q_0$ is an initial heat distribution
satisfying the initial conditions set forth in Section 3.2.
For each fixed positive integer $j$ we set $\Delta t = 2^{-j}$ and choose
$$\l K_j\r \colon \R^+ \to \KK, \quad Q_j \in \L_{2,loc}\big[\R^+, \, \L_2(\O)\big],
\quad T_j \in \L_{2,loc}\big[\R^+, \, \H^1(\O)\big]$$
by the following requirement and 
inductive procedure which alternates crystal changing steps with
heat flow steps.
\smallskip\noindent
{\bf 2.11.1} $\l K_j(0) \r = \l K_0\r$ and  $Q_j(0) = Q_0$.
\smallskip\noindent
{\bf 2.11.2} For each $k = 0, \, 1, \, 2, \, 3, \, \ldots$ and each $0 \le s < \Delta t$,
$\l K_j \big(k \Delta t + s\big)\r = \l K_j\big(k \Delta t\big)\r$.
\smallskip\noindent
{\bf 2.11.3} Suppose $k\in \{ 0, \, 1, \, 2, \, \ldots\}$ is
fixed and that
crystal position $K_j\big(k \Delta t\big)$,
heat distribution $Q_j\big(x, \, k \Delta t\big)$,
and temperature field $T_j\big(x, \, k \Delta  t\big)$
have already been constructed.
Assuming that functions $Q(x, \, t)$ and $T(x, \, t)$
defined on $\O \times [k \Delta t, \, (k+1) \Delta t]$ 
are a solution to the heat equation (see Theorem~5.1)
with fixed crystal $K_j\big(k \Delta t\big)$ and with 
initial heat distribution $Q_j\big(k\Delta t\big)$.
We set
$$Q_j\big(x, \, t\big) = Q\big(x, \, t\big)
\quad\hbox{if } k \Delta t \le t < (k+1)\Delta t, $$
$$T_j\big(x, \, t\big) = T\big(x, \, t\big)
\quad\hbox{if } k \Delta t \le t < (k+1)\Delta t, $$
$$P_j\big(x, \, (k+1)\Delta t\big)
= Q\big(x, \, (k+1)\Delta t\big)$$
\smallskip\noindent
{\bf 2.11.4} For each $k = 1, \, 2, \, 3, \, \ldots$ the pair
$\l K_j\big(k\Delta t\big)\r$, $Q_j\big(x, \, k \Delta t\big)$
give an $\EEE$ minimizer
for $P_j\big( k \Delta t\big)$ and $\Delta t$.  Also
$\displaystyle T_j\big(\cdot, \, k \Delta t\big) = c_{K_j(k
\Delta t)}^{-1} Q_j\big(\cdot, \, k \Delta t\big)$.

As additional notation we set
$$U_j(x, \, t) = {1 \over T_j(x, \, t)}$$
for each $j$, $x$, and $t$.  The functions $U_j$ are a convenience
in controlling
the behavior of our evolutions in regions where $T_j$ is small. 

We often write
$T_j(t) = T_j(\cdot, \,t )$,
$\nabla T_j(t) = \nabla T_j(\cdot, \,t )$,
$U_j(t) =U_j(\cdot, \, t)$,
$\nabla U_j(t) =\nabla  U_j(\cdot, \, t)$.

The existence of $\EEE$ minimizers is established in Chapter~4 and the existence
of heat flows is established in Chapter~5.  In Chapter~6 we
give conditions sufficient to guarantee that crystal positions
and temperature fields be close. The main theorems in Chapter~7
assert that, for a subsequence of the $j$'s, the crystals, heat distributions, and temperature
fields will converge strongly to limits which constitute
a solution to the heat equation with varying crystals.  The existence
of the Gibbs-Thomson relation is treated in Chapter~8.
\bigskip
{\bf 2.12 The total energy functions.}  For each positive integer $j$ we set
$$\EE_j \colon \R^+ \to \R^+, \quad \EE_j(t) = \EEE\big( \l K_j(t) \r, \, Q_j(t) \big)
\hbox{ for each } t.$$
Also, for convenience, we set
$$\EE_j(t^-) = \lim_{s \uparrow t} \EE_j(s)$$
for each $j$ and each positive~$t$.
Since one always has the option of doing nothing at the crystal changing steps
$\EE_j$ will not increase as a result of crystal changing.  We also show in
Section 5.2 that $\EE_j$ will not increase under any heat flow with fixed
crystal.  It follows that $\EE_j$ is nonincreasing as a function of $t$.
This provides an essential ingredient of our estimates.  
We set
$$\EE_0 =\EEE\big(\l K_0 \r, \, Q_0\big) = \EE_j(0)$$
for any $j$ since this number is independent of~$j$.
\bigskip
\centerline{\larger 3.  Initial conditions}
\bigskip
In this chapter we discuss some  of the  phenomena which can
arise in our mathematical model of crystal growth and state
initial conditions which preclude wildly oscillatory
solutions.  We show in Theorm~3.15 that there are
conditions under which, after a slight delay, a cold region will instantaneously
crystallize.  Such conditions are illustrated in Figure~3.
In particular, crystal evolutions $\l K_\ast(t) \r$ constructed
in the present model need not be continuous.
\bigskip
{\bf 3.1 Proposition (Some preliminary estimates about our
model).}
{\sl
\smallskip\noindent
(1) Suppose $Q_\star$ is a prescribed average heat
distribution, $0 \le \gamma \le 1$, and $\l K \r$ is a crystal
with $\LL (K)= \gamma\, |\O|.$
Then the bulk energy 
$$ \int_\O c_{{}_K}F(T)\, d \LL$$
is minimized if the temperature field is taken to be constant
with value $T(\gamma)$ satisfying the condition
$$ \big[\gamma\, c_{{}_s} + (1 - \gamma)
c_{{}_\ell}\big] T(\gamma) = Q_\star.$$
For this value of temperature, the total bulk energy equals
$$
\big[\gamma\, c_{{}_s} + (1 - \gamma) c_{{}_\ell}\big]
F\big(T(\gamma)\big)\, |\O|.
$$
\smallskip\noindent
(2) For each $0\le \gamma \le 1$ there is a crystal
$\l K(\gamma)\r $  with $\LL\big(K(\gamma)\big)= \gamma\, |\O|$
such that
$$\Phi\big(\partial \l K(\gamma) \r\big)= \inf\bigg\{\Phi
\big(\partial \l L \r \big)\colon \l L \r \hbox{ is a crystal
with } \LL (L) = \gamma\, |\O |\bigg\}.$$
For all sufficiently small positive $\gamma$,
any $K(\gamma)$ will be a Wulff shape for $\Phi$
(Wulff shapes in $\R^n$ are defined in {\rm [ATW 5.1]}; Wulff
shapes in our torus $\O$ have the obvious definition).
Similarly, for all $\gamma$ sufficiently close to~1, any $K(\gamma)$
will be a Wulff shape for the surface energy function
$\Phi^\ast(v) = \Phi(-v)$.
}
\bigskip
PROOF.
The uniform convexity of $F$ assumed in 2.9.3 implies that
transference of a small amount of heat from a region of higher
temperature to a region of lower temperature will
always strictly decrease total bulk energy (see the
calculations given in the proof of Proposition~A.2).  We infer
that only constant temperatures can (and indeed do) provide
minimal bulk energy for fixed crystal volume.
The value of the temperature
$T(\gamma)$ in~(1) is determined
by the prescribed total heat content in accordance with our
equation there, and the associated bulk energy in~(1) is also
determined by our equation there.

The existence of $\l K(\gamma)\r$'s asserted in~(2) follows
from the compactness theorem for integral currents together
with the lower semicontinuity of integrals of  convex
integrands.  As is well known, in $\R^n$  Wulff shapes uniquely
minimize surface energy for the volume enclosed. 
In order to verify our assertions about Wulff shapes, we 
first translate components of the support of $\l K(\gamma)\r$ 
in $\O$ if necessary to make
the support of $\l K(\gamma)\r$ connected.
We then  use
the constructions of Section~3.4 of~[ATW]
(as in the proof of conclusion~(1) of Theorem~4.5 below)
to verify the existence of a positive lower bound to
density ratios
$$\HH\big(\partial K  \cap \B^n(p, \, r)\big) \over r^{n-1}$$
associated with $p$'s in the support of $\partial \l K \r$ 
and small $r$'s.
We now infer from this lower density ratio bound that the
diameter of the support of $\partial \l K \r$ will be small if
the $n-1$ area of $\partial K$ is small and that this $n-1$
area will be small if the $n$~volume of $K$ is small.
We now use the fact that the simply connected covering space of
$\O$ is $\R^n$, etc.
\E
\bigskip
{\bf 3.2 Remarks.}
\smallskip\noindent
(1) For preassigned heat capacities $c_{{}_s}$ and $c_{{}_\ell}$
and preassigned average heat distribution $Q_\star$, the first
formula in 3.1(1) shows
$T(\gamma)$ to be an increasing function of $\gamma$, i.e. the
more of $\O$ that is frozen, the warmer the
the average temperature.
\smallskip\noindent
(2) For preassigned heat capacities $c_{{}_s}$ and $c_{{}_\ell}$,
preassigned $0 \le \gamma \le 1$,
and preassigned temperature $T(\gamma)$,
the first formula in 3.1(1) prescribes
the average temperature $Q_\star$.
If one preassigns $T(1) = T_\star$, then the lowest energy state
occurs only when $\O$ is completely frozen.  Similarly, if
one preassigns $T(0) = T_\star$ then the lowest energy state
occurs only when $\O$ is completely melted.
\smallskip\noindent
(3) It is not difficult to prescribe $F$, $c_{{}_s}$, 
$c_{{}_\ell}$, and $Q_\star$ so that 
$$c_{{}_s}F(T(1))= c_{{}_\ell}F(T(0)).$$
If $\Phi$ is then sufficiently large (when restricted to $\partial
\B^n(0, \, 1)$) then the total energy function
$$ \gamma \mapsto \Phi\big(\partial \l K(\gamma)\r \big) +
\big[\gamma\, c_{{}_s} + (1 - \gamma) c_{{}_\ell}\big]
F\big(T(\gamma)\big)\, |\O| $$
is minimized only when $\gamma=0$ or when $\gamma = 1$.
Following the terminology of 2.10 we see that
$\l K(0) \r$ and $Q_\star$ give an $\EEE$ minimizer for
$Q_\star$ and any value of $\Delta t$ and also
$\l K(1) \r$ and $Q_\star$ give an $\EEE$ minimizer for
$ Q_\star$ and any value of $\Delta t$.
If one were to construct approximating evolutions by the procedure
set forth in~2.11 with initial conditions $\l K_0 \r = \l
K(0)\r$ and $Q_0 = Q_\star$, then
$$\l K_j(t) \r = \cases{\l K(0)\r&\cr
\phantom{x}&\cr
\l K(1)\r&\cr}.$$
are both admissible choices for each $j$ and each $t$.  
\bigskip
{\bf 3.3 Controls on our evolution process.}
There are two main controls which we have on our evolution
process.  The first of these is the {\sl a priori} H\"older
estimates we obtain on our heat flow in the Monge-Kantorovich
distance; the essential estimate
is that set forth in Theorem~5.3, and this estimate depends
only on our bound $\EE_0$ for our initial total energy (2.12).
The second type of control we have control we have is one which
says (with the exception noted below) that if crystal
positions $K$ and $L$ are substantially different while heat
distributions $P$ and $Q$ are close, then one of
the associated temperature fields $c_K^{-1} P$ or $c_L^{-1}Q$
or the associated reciprocal temperature fields
$c_KP^{-1}$ or $c_LQ^{-1}$ must have large space
Dirichlet's integral; this estimate is
essentially set forth in Theorem~6.1, Corollary~6.2, and
Proposition~6.3.  The reason that Dirichlet's integrals are
controlled is that the rate of decrease of our Lyapunov
function $\EE_j(t)$ is bounded from below by the time
integral of these space Dirichlet's integrals; this is established
as Conclusion~(2) of Theorem~5.2.

The exceptional case when different $K$ and $L$ do not necessarily
cause large Dirichlet's integrals of temperature or reciprocal
temperatures is when $K$ is 
very small or when $K$ is nearly equal to $\O$ and
$L$ is nearly equal to $\O \sim K$.  
We therefore assume the following:
\bigskip
{\bf  3.4 Main assumption about initial conditions.}
We assume that in Section 2.12  our initial crystal $\l K_0\r$
and our initial heat distribution $Q_0$ are such that:
\smallskip\noindent
{\bf either} the volume of every crystal which can occur in any
approximating evolution is uniformly bounded away from zero 
\smallskip\noindent
{\bf or} the volume of every crystal which can occur in any
approximating evolution is uniformly bounded away from $|\O|$.
\bigskip
{\bf 3.5 Sufficient conditions that our main assumption about
initial conditions hold.}
Two conditions which together imply our main assumption are
the following (we use the terminology of~3.1 and~3.2):
\smallskip\noindent
(1) There is some number $0 < \gamma_0 < 1$ for which
$$
\eqalign{\Phi\big(\partial \l K(\gamma_0) \r\big)+
\big[\gamma_0\, c_{{}_s} + (1 - \gamma_0) c_{{}_\ell}\big]
&
F\big(T(\gamma_0)\big)\, |\O|
\cr &
< \sup\big\{
c_{{}_\ell} F\big(T(0)\big)\, |\O|, \, \,
c_{{}_s} F\big(T(1)\big)\, |\O|
\big\}.\cr}
$$
\smallskip\noindent
(2) Our initial crystal $\l K_0\r$ and initial heat
distribution $Q_0$ are chosen in 2.12 so that
$$\EE_0 = \EEE\big(\l K_0 \r, \, Q_0\big)
< \sup\big\{
c_{{}_\ell} F\big(T(0)\big)\, |\O|, \,\,
c_{{}_s} F\big(T(1)\big)\, |\O|
\big\}.
$$
Condition (1) says that it is possible in our model of crystal
growth to satisfy
condition~(2) while condition~(2) clearly implies
that our main assumption about initial conditions does indeed
hold.  A typical situation in which conditions~(1) and~(2) would 
hold would be that in which
(i) the ambient domain $|\O|$ is large; (ii) the initial
crystal volume $\LL(\K_0)$ is small;  and (iii) the
undercooling determined by $Q_0$ is large enough to permit
substantial crystal growth, but is not large enough to permit
freezing all of $\O$.
\bigskip
We now begin constructing an  example in which there is an
instantaneous appearance of a large crystal at positive time
from a pure liquid environment.  This shows, in particular, that
there can  be no general theorem guaranteeing continuous
crystal evolution under the conditions of the present model.
First we give a formula for the heat kernel under special
conditions and state well known facts about solutions to the heat
equation.  We do this to be able to justify claims about heat 
evolution we need in constructing our example.
\bigskip
{\bf 3.6 Solving the heat equation in $\O$ with no crystal
present.}  
In accordance with our definitions in 2.5 we say that 
$Q(x, \, t)$ and $T(x, \, t)$ defined on $\O\times [0, \, t_0]$ are a
solution to the heat equation with initial heat distribution $Q_a$ 
(and empty crystal $\l \emptyset \r$) provided
$$c_{{}_\ell} T = Q, \quad \dotq =
\hbox{div}\big(\kappa_{{}_\ell} \nabla T\big), \quad\hbox{and}
\quad Q(\cdot, \, 0) = Q_0.$$
Suppose that our heat capacitiy $c_{{}_\ell}$ equals one and
our heat conductivity matrix $\kappa_{{}_\ell}$ equals the
identity matrix.  Our heat equation then becomes
$$ \dott = \Delta  T \quad\hbox{with} \quad
T(\cdot, \, 0) = T_0 = Q_0.\eqno{(\ast)}$$
If our domain were $\R^n$ instead of $\O$ we could solve
$(\ast)$ by setting
$$T(x, \, t) = {1 \over \left(4 \pi t\right)^{n/2}}
\int_{\xi \in {\bf R}^n} T_0(\xi)\, \hbox{exp}\left(- {|x -
\xi|^2 \over 4t}\right)\, d \LL \xi$$
[RZ Problem 10, page 373].  If, additionally, 
$T_0(\xi + \rho_\ast z) = T_0(\xi)$ for each $\xi$
and each $z \in \Z^n$ then $(\ast)$ would become
$$T(x, \, t) = {1 \over \left(4 \pi t\right)^{n/2}}
\sum_{z \in {\bf Z}^n}\int_{\xi \in [0, \, \rho_\ast]^n}
T_0(\xi)\, \hbox{exp}\left(- {|x - (\rho_\ast z +
\xi)|^2 \over 4t}\right)\, d \LL \xi$$
We now fix a standard covering mapping $\Pi \colon \R^n
\to \O$ such that $\Pi(x + \rho_\ast z) = \Pi(x)$ for each 
$z \in \Z^n$.
We define a heat kernel
$Y \colon \O \times \R^+ \times \O$
by setting for each $(x, \, t, \, \xi) \in \O \times \R^+ \times \O$,
$$Y(x, \, t, \, \xi) = {1 \over \left(4 \pi t\right)^{n/2}}
\sum_{z \in {\bf Z^n}} \hbox{exp}\left(-{|z - (\rho_\ast z + 
\xi_\ast)|^2 \over 4t}\right);$$
here $x_\ast, \, \xi_\ast$ are any fixed points in $\R^n$ with
$\Pi(x_\ast ) =x$, $\Pi(\xi_\ast ) =\xi$.
In our original domain $\O$ we infer that the function
$$T(x, \, t) = \int_{\xi \in \O} T_0(\xi)\, Y(x, \, t,
\, \xi)\, d \LL \xi\eqno{(\ast \ast)}$$
solves the heat equation $(\ast)$ with initial
temperature~$T_0$.
\medskip
We state some well known properties of solutions to our heat
equation $(\ast)$ which follow from $(\ast \ast)$ and
which we will use in constructing our examples.
\smallskip\noindent
{\bf 3.6.1 Maximum and minimum principles.} If
$-\infty< A \le T_0(x) \le B < \infty$ for each $x$ in $\O$,
then
$-\infty< A \le T(x, \, t) \le B < \infty$ for each $x$ in $\O$
and each $t > 0$.
\smallskip\noindent
{\bf 3.6.2 Maximum principle for Lipschitz constants.}  Suppose
$T_0$ is smooth.  Then, for each $t > 0$,
$\lip\big(T(\cdot, t)\big) \le \lip\big(T_0\big)$, since
partial derivatives of $T(x, \, t)$ are also solutions to  the
heat equation.
\smallskip\noindent
{\bf 3.6.3 Uniform rate of dissipation.} For  each $\epsilon >
0$ and $\tau > 0$ there is $\delta > 0$ such that if
$$\int_\O |T_0|\, d \LL < \delta,$$
then $|T(x, \, t)| < \epsilon$ for each $x \in \O$ and each $t >
\tau$. 
\bigskip
{\bf 3.7 Proposition (An isoperimetric-intersection estimate).}
{\sl
Corresponding to each $0 < \lambda < 1$ there is $\epsilon > 0$
with the following property.  Suppose $R$ and $S$ are positive
numbers, $K_0 = \B^n(0, \, R)$,  and $K$ is any subset of $\R^n$
of finite perimeter
such that
\smallskip\noindent
(a) $\LL\big(K \tt K_0\big) < \epsilon R^n$,
\smallskip\noindent
(b) $\LL\big(K\big) = \alpha(n) S^n$ and
$\HH\big(\partial K\big) - n\alpha(n) S^{n-1} < \epsilon
S^{n-1}.$
\smallskip\noindent
Then, for each $0 < \delta < \epsilon$,
$${ \LL\bigg( K \cap K_0\cap \big\{ x \colon |x_n| < \delta
R \big\}\bigg)\over
\LL\bigg( K_0\cap \big\{ x \colon |x_n| < \delta R \big\}\bigg)}
> \lambda.$$
}
\bigskip
PROOF.  It clearly suffices to consider the case $R = 1$.
The isoperimetric inequality insures us that in case
$\HH\big(\partial K\big) = n\alpha(n) S^{n-1}$ then
$K$ will be a ball of radius $S$.  We use the terminology
of Appendix~C.1 and infer from Theorem~C.3 that if
$S$ is close to $1$ and if
$\HH\big(\partial K\big)$ is only slightly greater than
$n\alpha(n) S^{n-1}$ and if we write
$\partial \l K \r =
{\bf t}\big(\partial K, \, 1, \, \ast {\bf n}_{{}_K} \big)$
then the varifold
${\bf v}\big(\partial K, \, 1, \, \ast {\bf n}_{{}_K} \big)$
is weakly close to the varifold
${\bf v}\big(\partial K_0, \, 1, \, \ast {\bf n}_{{}_{K_0}} \big)$.
We set $\xi(x_1, \, \ldots, \, x_n) =
(x_1, \, \ldots, \, x_{n-1})$ and infer that for most
$z = (x_1, \, \ldots, \, x_{n-1})$ with
$r^2 = x_1^2 + \, \ldots\, + x_{n-1}^2 < 1$
the slice $\big< \l K \r, \, \xi, \, z\big>$ equals
$\l (z, h), \, (z, \, -k)\r$ for some $h$ and $k$ each close to
$\sqrt{1-r^2}$ [FH 4.3.8, page~444].
For such $z$'s, the set $\{z\} \times
[\delta, \, \delta]$ almost surely almost lies in $K$.  Our
assertion follows from Fubini's theorem.
\E
\bigskip
{\bf 3.8 Proposition (Isoperimetric inequalities).}
{\sl
There is a positive number $S_0$ and
an increasing function $\delta \colon \R^+\to \R^+$ with the
following property.  Suppose $0 < S \le S_0 \rho_\ast$
and $\l K \r$ is a
crystal in $\KK$ with $\LL(K) = \alpha(n) S^n$.  Then
\smallskip\noindent
(1) $\HH(\partial K) \ge n \alpha(n) S^{n-1}$ with equality if
and only if there is $p\in \O$ such that $\l K \r = \l \B^n(p,
\, S) \r$.
\smallskip\noindent
(2) If $\epsilon > 0$ and
 $\HH(\partial K) - n \alpha(n) S^{n-1} < \delta (\epsilon)\,
 S^{n-1}$, then
there is $p\in \O$ such that $\LL\big( K \tt \B^n(p,\, S)\big)
< \epsilon \, S^n$.
}
\bigskip
PROOF.  The first assertion follows from the remarks about
Wulff shapes in Proposition~3.1(2); see also [ATW 3.1.7].
The second assertion
follows from the first assertion and compactness theorems for
integral currents; see, for example, [ATW 3.1.5]
\E
\bigskip
{\bf 3.9 Assumptions and terminology.}
We specify certain temperatures, bulk energy changes,
a radius, and $\rho_\ast$.  We have in mind regions
``$A$, $B$, $C$'' (to be specified below) containing
liquid at temperatures $\tau_A$, $\tau_B$, $\tau_C$ which
change to temperatures $\tau_{AA}$, $\tau_{BB}$, 
$\tau_{CC}$ upon crystallization with changes
$\Delta F_A$, $\Delta F_B$, $\Delta F_C$ in bulk energy
densities.
\smallskip \noindent
(1) We assume that our liquid heat capacity $c_{{}_\ell}$
equals~1 and that our solid heat capacity $c_{{}_s}$ is a fixed
number less than one.
\smallskip \noindent
(2) We assume that our liquid heat conductivity 
$\kappa_{{}_\ell}$ is the identity matrix. 
\smallskip \noindent
(3) We assume that our bulk energy density function $F$ and
our equilibrium freezing temperature~$T_\star$ are fixed.
In accordance with 2.9.2, $F(T_\star) =0$.  In our terminology
below, $\tau_{BB} = \tau_C = T_\star$.
\smallskip \noindent
(4) We set $\tau_B = c_{{}_s} T_\star / c_{{}_\ell}
=  c_{{}_s} T_\star< T_\star $
so that liquid at temperature $\tau_B$ will change
to temperature $\tau_{BB} = T_\star$ on solidification.
The negative number 
$\Delta F_B = c_{{}_s}F(\tau_{BB}) - c_{{}_\ell}F(\tau_B)
= - F(\tau_B)$ is the change (decrease) in bulk energy density
when liquid at temperature $\tau_B$ solidifies.
\smallskip\noindent
(4) We set $\tau_C = T_\star$
so that liquid at temperature $\tau_C$ will change to
temperature $\tau_{CC} = c_{{}_\ell} \tau_C / c_{{}_s}
= \tau_C / c_{{}_s}$ on solidification.  The positive number
$\Delta F_C = c_{{}_s} F(\tau_{CC}) - c_{{}_\ell}F(T_C)=
 c_{{}_s} F(\tau_{CC})$ is the change (increase) in bulk energy density
when liquid at temperature $\tau_C$ solidifies.
\smallskip \noindent
(5) We denote by $\tau_A$ a temperature much greater than $\tau_{CC}$
and also greater than $b_\ast$ (see 2.9.4)
whose value will be specified later and set
$\tau_{AA} = c_{{}_\ell} \tau_A /c_{{}_s} = \tau_A /c_{{}_s} > \tau_A$
so that liquid at temperature $\tau_A $ will change
to temperature $\tau_{AA}$ on solidification.  The positive
number
$$\Delta F_A = c_{{}_s}F(\tau_{AA}) - c_{{}_\ell}F(\tau_A)
=c_{{}_s} \tau^2_{AA} - c_{{}_\ell} \tau_A^2 = 
c_{{}_s}\left({ \tau_A\over c_{{}_s}}\right)^2 - \tau_A^2 =
{1-c_{{}_s} \over c_{{}_s}} \tau_A^2$$
is the
change (increase) in bulk  energy density when liquid at
temperature $\tau_A$ solidifies.
\smallskip \noindent
(6) We fix our surface energy $\Phi$ to be surface area by
requiring $\Phi(v) = |v|$ for each $v$.
\smallskip \noindent
(7) We fix a positive radius $R_0$ by the requirement that
$\Delta F_B\, \alpha(n)\, R_0^n =n\,\alpha(n)\,R_0^{n-1}.$
It follows that if $p \in \O$ and
$$Q_0 = \bigg(\tau_B \upchi_{{}_{{\bf B}^n(p, \, R_0)}} +
\tau_C\upchi_{{}_{\O \sim {\bf B}^n(p, \, R_0)}}\bigg)$$
then, for $p \in \O$,
$$\EEE\big(\l \emptyset \r, \,Q_0\big)= 
\EEE\big(\l \B^n(p, \, R_0) \r, \,Q_0\big).$$ 
\smallskip \noindent
(8) We let $S_0$ have the meaning of Proposition~3.8 and pick
$\rho_\ast = 2 R_0 / S_0$
\bigskip
The Monge-Kantorovich term in our minimization criterion used
in constructing approximating evolutions adds a subtlety to our
estimates.  The following proposition helps control the  
influence of this term.
\bigskip
{\bf 3.10 Proposition (An estimate relating bulk energy change
and Monge-Kantorovich distance).}
{\sl
Suppose
\smallskip\noindent
(a) $\l K \r$  is a crystal in $\O$.
\smallskip\noindent
(b) $P$ and $Q$ are heat distributions in $\QQ$
\smallskip\noindent
(c) $R = P/c_{{}_K}$ is the temperature field associated
with heat distribution $P$ with crystal $\l K \r$ present.
\smallskip\noindent
(d) $S = Q/c_{{}_K}$ is the temperature field associated
with heat distribution $Q$ with crystal $\l K \r$ present.
\smallskip\noindent
(e) $T = P/c_{{}_\ell}\le R$ is the temperature field associated
with heat distribution $P$ with no crystal present.
\smallskip\noindent
(f) $\lip(T) = \lip(P)/c_{{}\ell} < \infty$.
\smallskip\noindent
(g) $0 < T_1 = \inf_\O P/c_{{}_\ell} = \inf_\O T
\le \sup_\O R \le \sup_\O P/c_{{}_s} = T_2 < \infty.$
\smallskip\noindent
(h) $\Gamma(P) = \sup_{T_1 \le U \le T_2} F''(U)< \infty$.
\smallskip\noindent
Then  the savings in total energy 
$$
\EEE\big(\l \emptyset\r, \, P\big) - \EEE\big(\l K \r, \, Q\big)$$
from initial empty crystal and heat distribution $P$ to
crystal $\l K\r$ and heat distribution $Q$ does not exceed
$$-\Phi\big(\partial \l K \r \big) +
\int_K c_{{}_\ell} F(T) - c_{{}_s}F(R) \, d \LL 
+ \Gamma(P)\, \lip(T)\, \| P-Q\|_\ast
+ \Gamma(P)\, \big(T_2 - T_1\big) \int_K P \, d \LL. $$
}
\bigskip
PROOF. Since $F$ is convex,
$F(S)\ge F(R) + F'(R)\,(S-R)$
so that
$$\eqalign{c_{{}_K}F(R) - c_{{}_K}F(S)  
& \le
c_{{}_K}F'(R)\, (R-S)
\cr &
=
F'(R) (P-Q)
\cr &
=
F'(T)\,(P-Q) + \big(F'(R) - F'(T)\big)\, (P-Q)
\cr &
=
F'(T)\,(P-Q) + F''(U)\,(R-T)\, (P-Q)
\cr}$$
for some $T \le U \le R$.
Hence
$$\eqalign{\EEE\big(\l K \r, \, P\big) - \EEE\big(\l K \r, \, Q\big)
&
=
\int_\O c_{{}_K} F(R) - c_{{}_K}F(S) \, d \LL -
\cr &
\le
\int_\O\big(F' \circ T\big) \, (P - Q) \, d \LL
+
\Gamma(P)\int_K (R - T) ( P- Q) \, d \LL
\cr &
\le
\lip\big(F' \circ T\big) \| P-Q\|_\ast
+ \Gamma(P)\, \big(T_2 - T_1\big) \int_K P \, d \LL
.}$$
Hence
$$\eqalign{\EEE\big(\l \emptyset \r, \, P\big)
- \EEE\big(\l K \r, \, Q\big)
&
=
\EEE\big(\l \emptyset \r, \, P\big)- \EEE\big(\l K \r, \, P\big)
+\EEE\big(\l K \r, \, P\big) - \EEE\big(\l K \r, \, Q\big)
\cr &
\le
-\Phi\big(\partial \l K \r \big) +
\int_\O c_{{}_\ell} F(T) - c_{{}_K}F(R) \, d \LL 
\cr &
\phantom{XXX}
+ \Gamma(P)\, \lip(T)\, \| P-Q\|_\ast
+ \Gamma(P)\, \big(T_2 - T_1\big) \int_K P \, d \LL
.\cr}$$
\E
\bigskip
{\bf 3.11 Assumptions and terminology.} 
\smallskip\noindent
(1)  We assume that our initial crystal $\l K_0\r$ is the zero
current corresponding to the empty set.  We denote by $T_0 =
Q_0$ our initial temperature fields and heat distributions
which we assume to be positive smooth functions.
\smallskip\noindent
(2) For each $j$ we denote by $\Delta t_j = 2^{-j}$ the time
step of the  $j$-th approximating evolution.
\smallskip\noindent
(3) For each $j$ and time $t$ we denote by
$\l K_j(t)\r$,  $Q_j(t)$, $T_j(t)$ the crystals, heat
distributions, and temperature fields at time $t$ in a $j$-th
approximating evolution constructed as in 2.11
Additionally for each positive integer~$k$,
$$P_j\big(k\,\Delta t_j\big)= \lim_{t \uparrow k\,\Delta
t_j} Q_j(t);$$
compare 2.11.13

\bigskip
{\bf 3.12 Corollary (The evolution when no crystal
is present is pure heat flow).}
{\sl
There is a positive integer $J$  with the following property.
Suppose $j \ge J$ and there is a positive time $t_0$
such that $K_j(t)= \emptyset$
for $0 \le t \le t_0$.
\smallskip\noindent
(1) If $k$ is a positive integer such that
$k \, \Delta t_j\le t_0$ then
$$Q_j\big(k \Delta t_j\big) = P_j\big(k\, \Delta t_j\big).$$
\smallskip\noindent
(2) The restriction of the function $Q_j(x, \, t)= T_j(x, \, t)$
to the time interval $[0, \, t_0]$ is a 
solution to the heat equation (with zero crystal) with 
initial
heat distribution $Q_0$ and final heat distribution~$Q_j(t_0)$.
}
\bigskip
PROOF. 
Conclusion~(1) clearly implies Conclusion~(2) since we are
solving the ordinary heat equation in~$\O$.
We now verify Conclusion~(1).  Suppose then $t_1= k \Delta
t_j$.  Proposition~3.10 above (with $K = \emptyset$ there) shows
that possible decreases in bulk energy from replacing
$P_j\big(k\, \Delta t_j\big)$ by $Q_j\big(k \Delta t_j\big)$
are dominated by $\| Q_j\big(k \Delta t_j\big) 
-P_j\big(k\, \Delta t_j\big)\|_\ast$ provided one can bound
temperatures above and below and also  bound the Lipschitz
constant of temperatures.  Such bounds hold at time $t = 0$
by assumption~3.11(1).  In accordance with 3.9(1)(2) together
with 3.6.1 and 3.6.2, so long as no crystal is present, the
bounds which hold for
$Q_j\big((k-1)\,\Delta t_j\big)$ also hold for
$P_j\big(k\, \Delta t_j\big)$.
In view of the factor $1/\Delta t_j^\alpha$ in the $\EEE$
minimizing equation given in 2.10, it is always better to
take $Q_j\big(k \Delta t_j\big)$ equal 
to $P_j\big(k\, \Delta t_j\big)$ when $\Delta t_j$ is small,
i.e. $j$ is large.
This gives the assertion of the Conclusion~(2).
\E
\bigskip
{\bf 3.13 Corollary (A lower bound on the size of first
crystallizations).}
{\sl 
There are a positive integer $J$ and a positive  number $\eta$
with the following  property.  Suppose $j\ge J$ and
$k$ is a positive integer such that
$\l K_{j}\big(\ell\,  \Delta t_j \big)\r = \l \emptyset\r$
for each $\ell < k$ while
$\l K_j\big(k \, \Delta t_j \big)\r \not= \l \emptyset\r$.
Then $\LL(K_j) \ge \eta$.
}
\bigskip
PROOF.  From 2.10 and 2.11 we infer 
$$\EEE\bigg(\l \emptyset \r, \, P_j(k \, \Delta t_j)\bigg)
- \EEE\bigg(\l K_j (k\, \Delta t_j) \r, \, Q_j(k \, \Delta t_j)\bigg)
- \Delta t_j^{-\alpha} \, \| P_j(k\, \Delta t_j)- Q_j(k\, 
\Delta t_j)\|_\ast \ge 0.$$
We infer from assumptions~3.9(1)(2) together
with 3.6.1 and 3.6.2 and Corollary~3.12 
that upper and lower bounds and Lipschitz bounds which hold
for $Q_0$ also hold for
$P_j\big(k\, \Delta t_j\big)$.
If $j$ is  large then $\Delta t_j$ will be small so that,
in the terminology of Proposition~3.10,
$$\Delta t_j^{-\alpha} \, \| P_j(k\, \Delta t_j)- Q_j(k\, 
\Delta t_j)\|_\ast 
\phantom{XXXXXXXXXXXXXXXXXXXXXXXXXX}$$
$$\phantom{XXX}> 
\Gamma\bigg(P_j(k\, \Delta t_j)\bigg)
\, \lip\bigg(P_j(k \, \Delta t_j)\bigg)
 \| P_j(k\, \Delta t_j)- Q_j(k\, \Delta t_j)\|_\ast.$$
In view of Proposition~3.10 and the isoperimetric inequality 
of Proposition~3.8 ($\rho_\ast$ was fixed in 3.9(8)),
this can happen only if $\LL\big(K_j(k\, \Delta t_j)\big)$ is large.
\bigskip
{\bf 3.14 Corollary (A total energy estimate on first
crystallizations).}
{\sl
Suppose $j(1) < j(2) < j(3) < \ldots$ and 
$k(1) < k(2) < k(3) < \ldots$ are positive integers 
and $t(i) = k(j(i))\, \Delta t_{j(i)}$ for each $i$
such that
$\l K_{j(i)}\big(t \big)\r = \l \emptyset\r$
if $t < t(i)$ while
$\l K_{j(i)}\big(t(i)\big)\r \not= \l \emptyset\r$. 
Suppose also $\sup_it(i) < \infty$.  
Then there is a subsequence 
$\ell(1), \, \ell(2), \, \ell(3), \, \ldots$ of $1, \, 2, \, 3,
\, \ldots$ together with a time $\tau$, a crystal
$\l K^0\r$ in $\KK$, and a
heat distribution $Q^0$ in $\QQ$ with associated temperature
field $T^0= Q_0/c_{{}_\ell} = Q^0$ such that
\smallskip\noindent
(1) $\tau = \lim_{\ell \to \infty} t(i(\ell))$.
\smallskip\noindent
(2) $\lim_{\ell \to \infty} \LL\big( K_{j(i(\ell))}\tt K^0 \big)
= 0$.
\smallskip\noindent
(3) $T_{j(i(\ell))}(t(i(\ell))) \h T^0$ and
$P_{j(i(\ell))}(t(i(\ell))) \h P^0$ and
$Q_{j(i(\ell))}(t(i(\ell))) \h P^0$ 
in $\L_2(\O)$ as $\ell \to \infty$.
\smallskip\noindent
(4) If $T(x, \, t)$ is the solution to the  heat equation
with initial temperature field~$T_0$ given in section 3.6, then
$P^0 = T^0 = T(\cdot, \, \tau)$.
\smallskip\noindent
(5) $\LL(K^0)> 0$.
\smallskip\noindent
(6) In terms of total energy, it is not disadvantageous to
crystallize $\l K^0 \r$ from pure liquid
in the temperature field $T(\cdot, \, \tau)$ with no change
in  the  underlying heat distribution, i.e.
$$\EEE\big(\l \emptyset \r, \, P^0\big)
-\EEE\big(\l K^0 \r, \, P^0\big) \ge 0.$$
}
PROOF.
Conclusion (1) is immediate.  Conclusion~(2) follows from 
compactness theorems for integral currents ([ATW 3.1.5] for
example); we infer further from the lower semicontinuity of
integrals of convex integrands under flat convergence
that $\Phi\big(\partial \l K \r\big)
\le \lim\inf_{\ell \to \infty}\Phi\big(\partial
\l K_{j(i(\ell))}(t(i(\ell))\r \big)$.
Conclusion (3) follows from Appendix~B.8.
Conclusion~(4) follows from Corollary~3.12.
Conclusion~(5) follows from Corollary~3.13.
According to 2.10 and 2.11 we have for each $i$,
$$\EEE\bigg( \l \emptyset \r, \, P_{j(i)}(t(i))\bigg)
-\EEE\bigg( \l K_{j(i)}(t(i)) \r, \, Q_{j(i)}(t(i))\bigg)
\ge \Delta t_{j(i)}^{-\alpha} \, \| P_{j(i)} - Q_{j(i)}
\|_\ast.$$
We infer from Appendix~B.4 that
$$\EEE\big( \l K^0 \r, \, Q^0 \big) \le \lim\inf_{\ell \to \infty}
\EEE\big( \l K_{j(i(\ell))} \r \big).$$
We then infer from Appendix~B.9 that $P^0 = Q^0$.  This gives
Conclusion~(6).
\E
\bigskip
{\bf 3.15 Assumptions and terminology.}
\smallskip\noindent
(1) $\Psi_1 \colon \R^n \to \R^+$ is a radially symmetric
smoothing function having support in $\B^n(0, \, 1)$ and
$\Psi_\epsilon(x) = \epsilon^{-n} \Psi(x/\epsilon)$ for each
$x$ and $\epsilon$.
\smallskip\noindent
(2) $p$ is a point in $\O$ and $v$ is a unit vector in $\R^n$.
\smallskip\noindent
(4) For $R_0 \le R < 2 R_0$ and $0 < \delta < 1/2$, 
$$B(R) = \B^n(p, \, R)$$
is a
closed ball, 
$$C(R) = \O \sim B(R)$$
is the complement of $B(R)$ in $\O$, and
$$A(R, \, \delta) = B(R) \cap \{x \colon |(x-p) \bullet v|
< \delta R_0\}$$
is a thicked disk in the ball $B(R)$ 
analogous to that of Proposition~3.7.
\smallskip \noindent
(5) Corresponding to radius $R_0 < R < 2R_0$,
temperature $\tau_A$, thickness parameter $\delta$, and
smoothing radius $\epsilon$ we
associate the initial heat distribution and temperature field
$$T_0[R, \,  \delta, \, \tau_A, \, \epsilon]
=
Q_0[R, \,  \delta, \, \tau_A, \, \epsilon]
= \Psi_\epsilon \,\ast\,\bigg[
\tau_A \upchi_{{}_{A(R, \, \delta)}}
+\tau_B \upchi_{{}_{B(R)}}
+\tau_C \upchi_{{}_{C(R)}}\bigg].$$
\bigskip
{\bf 3.16 Theorem (A large crystal which appears
instantaneously at positive time).}
{\sl
There exists a choice of 
radius $R_0 < R < 2R_0$,
temperature $\tau_A$, thickness parameter $\delta$, and
smoothing radius $\epsilon$
such that with the initial heat distribution and temperature field
$$T_0[R, \,  \delta, \, \tau_A, \, \epsilon]
=Q_0[R, \,  \delta, \, \tau_A, \, \epsilon]$$
there will be positive times $0 < t_\ast < t^\ast$ with the
following property:  For all sufficiently large integers $j$
there will be a time $t_\ast \le t_j \le t^\ast$ such  that
\smallskip\noindent
(1) $\l K_j(t) \r = \l \emptyset \r$ for all $0 \le t < t_j$.
\smallskip\noindent
(2) $\LL\big( K_j(t) \tt B(R_0)\big) \ge \LL\big(B(R_0)\big)$
for all $t_j \le t < t^\ast$.
}
\bigskip
PROOF.  
For $i = 1, \, 2, \, 3, \, \ldots$, we set $R_i = (1 + 2^{-i})R_0$ 
and
$$Q_i = \bigg[ \tau_B \upchi_{{}_{B(R_i)}}
+\tau_C \upchi_{{}_{C(R_i)}}\bigg].$$

{\bf Step 1.}  
For each $i$ there is a crystal $\l K_i\r$ such that
$$\EEE\big(\l \emptyset \r, \, Q_i\big) -
\EEE\big(\l K_i \r, \, Q_i\big)> 0$$
Suppose $\l K_1\r, \, \l K_2\r, \, \l K_3\r , \, \ldots$
are crystals for which
$$\EEE\big(\l \emptyset \r, \, Q_i\big) -
\EEE\big(\l K_i \r, \, Q_i\big)\ge 0.$$
Then  we infer from our construction and Proposition~3.8 that
$$\LL\big( K_i \tt B(R_0)\big) \to 0 \quad \hbox{and} \quad
\HH\big(\partial K_i \big) - n\, \alpha(n)\, R_0^{n-1} \to 0$$
as $i \to \infty$ and we infer  from Proposition~3.7 that
$${\LL\big(K_i \cap A(R_0, \, \delta \big)\over
\LL\big(A(R_0, \, \delta \big)} \to 1$$
uniformly in $\delta$ as $i \to \infty$.
\bigskip
{\bf Step 2.} There exist positive numbers
$\epsilon_1, \, \epsilon_2, \, \epsilon_3, \ldots$ and
$\sigma_1, \, \sigma_2, \, \sigma_3, \ldots$ 
with the following properties.  Suppose
$$Q_{i,\epsilon} = \Psi_{\epsilon_i} \ast Q_i \pm \sigma_i.$$
For each $i$ there is a crystal $\l K_i\r$ such that
$$\EEE\big(\l \emptyset \r, \, Q_{i, \epsilon}\big) -
\EEE\big(\l K_i \r, \, Q_{i, \epsilon}\big)> 0$$
Suppose $\l K_1\r, \, \l K_2\r, \, \l K_3\r , \, \ldots$
are crystals for which
$$\EEE\big(\l \emptyset \r, \, Q_{i, \epsilon}\big) -
\EEE\big(\l K_i \r, \, Q_{i, \epsilon}\big)\ge 0.$$
Then
$$\LL\big( K_i \tt B(R_0)\big) \to 0 \quad \hbox{and} \quad
\HH\big(\partial K_i \big) - n\, \alpha(n)\, R_0^{n-1} \to 0$$
as $i \to \infty$ and
$${\LL\big(K_i \cap A(R_0, \, \delta \big)\over
\LL\big(A(R_0, \, \delta \big)} \to 1$$
uniformly in $\delta$ as $i \to \infty$.
\bigskip
{\bf Step 3.} There exist positive times $0 < t_\ast< t^\ast$
together with positive numbers 
$\epsilon_1, \, \epsilon_2, \, \epsilon_3, \ldots$ and
$\sigma_1, \, \sigma_2, \, \sigma_3, \ldots$ and
$\delta_1, \, \delta_2, \, \delta_3, \ldots$ nd
$\tau_1, \, \tau_2, \, \tau_3, \ldots$ 
such that with the initial heat distribution and temperature field
$$T_0[R_i, \,  \delta_i, \, \tau_{A_i}, \, \epsilon_i]
=Q_0[R_i, \,  \delta_i, \, \tau_{A_i}, \, \epsilon_i]$$
there will be positive times $0 < t_\ast < t^\ast$ with the
following property:  For all sufficiently large integers $j$
there will be a time $t_\ast \le t_j \le t^\ast$ such  that
\smallskip\noindent
(1) $\l K_j(t) \r = \l \emptyset \r$ for all $0 \le t < t_j$.
\smallskip\noindent
(2) $\LL\big( K_j(t) \tt B(R_0)\big) \ge \LL|big(B(R_0)\big)$
for all $t_j \le t < t^\ast$.
The main idea is to choose the $\delta_i$'s very small and the
$\tau_{A_i}$'s very large in such a way that
$\LL(A(R_i, \delta_i)) \tau_{A_i}$ is very small while
$\LL(A(R_i, \delta_i)) \tau_{A_i}^2 $ is very large.
We infer from Step~2 and Corollary 3.12 and 3.6 and especially
Proposition~3.14 that no
crystallization will occur until the high temperatures in
the $A(R_i, \, \delta_i)$'s  will have had time to diffuse away.
According to 3.6 there is a positive lower bound to this time
provided the $\delta_i$'s and $\tau_{A_i}$'s are fixed for
large~$i$'s.  According to 3.6.3 the high temperatures  in
the $A(R_i,\, \delta_i)$'s will dissipate in uniform time.
We infer from Step~2 that crystallization will have occurred by
then and that it will have been large.
\E
\bigskip
\centerline{\larger 4. Existence and structure of $\EEE$ minimizers}
\bigskip
In this chapter we use existence and regularity theorems from
geometric measure theory to show that the variational
minimizations required to construct  our approximating
evolutions can be performed.  The resulting temperature fields
are Lipschitz continuous, and the Gibbs-Thomson relation
between curvature and temperature holds exactly along the
crystal interface.  In two and three dimensional spaces, the
crystal interface is a smooth hypersurface in  case $\Phi$ is
smooth and elliptic.
\bigskip
{\bf 4.1 Theorem (Existence of $\EEE$ minimizers).}
{\sl
Corresponding to each heat distribution $P \in \QQ$ and
each time interval $\Delta t >0$ there exist a crystal
$\l K \r \in \KK$ and a heat distribution $Q \in \QQ$
which give an $\EEE$ minimizer for $P$ and $\Delta t$.
}
\bigskip
PROOF.  Suppose
$\l K_1\r, \, \l K_2\r, \, \l K_3\r, \, \ldots \in \KK$ and 
$ Q_1, \,  Q_2, \,  Q_3, \, \ldots \in \QQ$ such that
$$\lim_i\bigg[ \EEE\big(\l K_i \r, \, Q_i\big) + 
\Delta t^{-\alpha}\| Q_i - P \|_\ast\bigg]
= \inf \bigg\{
\EEE\big(\l K \r, \, Q\big) + \Delta t^{-\alpha}\| Q - P \|_\ast \colon
\l K \r \in \KK, \, Q \in \QQ\bigg\}.$$
We set $T_i = c_{K_i}^{-1} Q_{i}$ for each $i$ as the
temperature field associated to $Q_i$ and $\l K_i \r$ so that
$$ \EEE\big(\l K_i \r, \, Q_i\big) + \Delta t^{-\alpha}
\| Q_i - P \|_\ast
= \Phi\big( \partial \l K_i \r \big) + \int_\O c_{K_i} F\left(
T_i\right)\, d \LL + \Delta t^{-\alpha} \| Q_i - P \|_\ast.$$
The compactness theorem for integral currents  guarantees the
existence of a subsequence $j(1), \, j(2), \, j(3),\, \ldots$
of $1, \, 2, \, 3, \, \ldots$ and $\l K \r \in \KK$ such that
$\LL(K_{j(i)} \tt K) \to 0$ as $i \to \infty$ and
$$\Phi\big(\partial \l K \r \big) \le
\lim \inf_i\Phi\big(\partial \l K_{j(i)} \r \big).$$
Corollary~B.6 guarantees the existence of a further
subsequence $k(1), \, k(2), \, k(3),\, \ldots$
of $j(1), \, j(2), \, j(3),\, \ldots$ and $Q \in \QQ$ such that
$$\| Q_{j(i)} - Q \|_\ast \to 0\quad \hbox{ and } \quad
\| Q_{j(i)} - P \|_\ast \to \| Q - P \|_\ast$$
as $i \to \infty$.
We then apply the convergence criteria given in~B.3
together with the  lower semicontinuity criterion given
in~Proposition~B.4 
(with $g_i$, $f_i$, $G$ there replaced by
$c_{K_{j(i)}}$, $T_{j(i)}$, $F$ respectively in the
present context) to infer
the existence of a final subsequence
$\ell(1), \, \ell(2), \, \ell(3),\, \ldots$
of $k(1), \, k(2), \, k(3),\, \ldots$ and limit temperature
field $T$ such that $T_{\ell(i)} \h  T$ as $i \to \infty$ (weak
convergence in $\L_2(\O)$) and
$$\int_\O c_K F(T)\, d \LL \le
\lim\inf_i \int_\O c_{K_{\ell(i)}} F(T_{\ell(i)})\, d \LL.$$ 
We use the convergence criteria in~ B.3 to infer that
$Q_{\ell(i)} = c_{K_{\ell(i)}} T_{\ell(i)} \h 
Q = c_K T$ as $i \to \infty$.
We do not assert any special convergence of the functions
$1/T_{\ell(i)}$ to $1 / T$.  \E
\bigskip
{\bf 4.2 Theorem (A priori estimates on temperature fields 
for $\EEE$ minimizers).}
{\sl
Suppose crystal
$\l K \r \in \KK$ and  heat distribution $Q \in \QQ$
give an $\EEE$ minimizer for some heat distribution
$P$ and time interval $\Delta t$.
Suppose also $T= c_K^{-1}Q$ is the temperature field
associated with $Q$ and $\l K\r$.  Then we can redefine $T$ on a set
of $\LL$ measure zero if necessary in order that the following
is true:
\smallskip \noindent
(1) For each $p$ in $\O$,
$$\Gamma(K,\,Q) \, \Delta t^{n \alpha / (3n+2)}
\le T(p) \le {Q_\ast|\O| \over c_{{}_s}} +
\left({n^{1/ 2} \rho_\ast \over 2 c_\ast}\right)
{1 \over \Delta t^\alpha};$$
here the constant
$$\Gamma(K, \, Q) =\left( {\alpha(n) \over
2^{(3n+2)/ 3} C_\ast^n \left[ \EEE\big(\l K\r, \, Q\big) + {\O
/ a_\ast^2}\right]}\right)^{1 \over 3n+2}$$
is bounded from below once
$\EEE\big(\l K\r, \, Q\big)$
is bounded from above.
\smallskip \noindent
(2) For each $p$ and $q$ in $\O$,
$$\big|F'(T(p)) - F'(T(q)) \big| \le {|p-q| \over \Delta
t^\alpha}, $$
$$\left| {1 \over T(p)^3} - {1 \over T(q)^3}\right| \le
 {C_\ast |p-q| \over  \Delta t^\alpha},$$
and
$$\big| T(p) - T(q) \big| \le {|p-q| \over c_\ast \Delta
t^\alpha};$$
in particular, $T$ is Lipschitz continuous.
}
\bigskip
PROOF. 
Since we are free to redefine $T$ on a set of measure zero
if we like, it is sufficient to verify our Lipschitz relation
between points $p$ and $q$ in $\O$ which are close and which
are Lebesgue points of both $T$ and $c_K$ [FH 2.9.9].  Suppose
$p$ and $q$ are such points with $T(p) < T(q)$ and set $v=q-p$
(with the obvious meaning). 
For arbitrary (but fixed) $0 < \epsilon << |v|/3$ 
we choose a
set $A$ in $\B^n(p, \, \epsilon)$ with $\LL(A) > 0$ such that
for each $x \in A$, the following conditions hold:
$$c_K(x) = c_K(p), \qquad 
c_K(x+v) = c_K(q),$$
$$
\big|Q(x) - Q(p)\big| < \epsilon, \qquad
\big|Q(x+v) - Q(q)\big| < \epsilon, \,$$
$$
\big|T(x) - T(p)\big| < \epsilon, \qquad
\big|T(x+v) - T(q)\big| < \epsilon.$$
For arbitrary (but fixed) $0 < \Delta Q < \epsilon c_{{}_s}$ we
consider a comparison heat distribution $Q^\ast \in \QQ$
given by setting
$$Q^\ast(x)= \cases{ Q(x)& if $x \not\in A \cup (\{v\} + A)$\cr
\phantom{a}&\cr
		    Q(x) + \Delta Q & if $x \in A$\cr 
\phantom{a}&\cr
		    Q(x) - \Delta Q & if $x \in \{v\} + A$.\cr }$$
We also set $\Delta T(p) = c_K(p)^{-1}\Delta Q$  
and $\Delta T(q) = c_K(q)^{-1}\Delta Q$.
The assumed minimality of $\l K \r$ and $Q$ implies that
$$\EEE\big(\l K \r, \, Q\big)  +
{1 \over \Delta t^{\alpha}}\|Q - P \|_\ast\le
\EEE\big(\l K \r, \, Q^\ast\big)+ {1 \over \Delta t^{\alpha}} \|Q^\ast
- P\|_\ast.$$
We estimate
$$\eqalign{
\int_\O c_K F\big(c_K^{-1}Q\big) \, d \LL - 
\int_\O c_K F\big(c_K^{-1}Q^\ast\big) \, d \LL 
& \le 
{1 \over \Delta t^\alpha} \|Q^\ast - P \|_\ast
- {1 \over \Delta t^\alpha} \| Q - P \|_\ast 
\cr &
\le {1 \over \Delta t^\alpha} \|Q^\ast - Q \|_\ast 
\cr &
\le{1 \over \Delta t^\alpha} \LL(A)\, \Delta Q\, |v|.\cr}$$
The extreme inequality will be used to prove our assertions.
We use the Mean Value Theorem in the following sequence
of equalities to
conclude, for each $x \in A$, the existence of
temperatures
$$T(x) < T^\ast(x) < T(x) + \Delta T(p) < T(x) + \epsilon, $$
$$ T(x+v) - \epsilon < T(x+v)- \Delta T(q) < T^\ast(x+v) < T(x+v)$$
for which the following is true.
$$\eqalign{
\int_\O c_KF(c_K^{-1} Q) \, d \LL
&
- \int_\O c_KF\big(c_K^{-1} Q^\ast\big) \, d \LL \cr
&
= \int_{x \in A} c_K(p)F\big(T(x)\big)
- c_K(p)F\big(T(x) + \Delta T(p)\big) \, d \LL x
\cr & \phantom{XX}
+ \int_{x \in A} c_K(q)F\big(T(x+v)\big) 
\cr & \phantom{XXXXXXXXXXXX}
- c_K(q)F\big(T(x+v)- \Delta T(q)\big) \, d \LL x \cr
\cr}
$$
$$\eqalign{
\phantom{\int_\O c_KF(c_K^{-1} Q) \, d \LL}
&
= \int_{x \in A} c_K(p)F'\big(T^\ast(x)\big)(- \Delta T(p)) \, d \LL x
\cr & \phantom{XX}
+ \int_{x \in A} c_K(q)F'\big(T^\ast(x+v)\big)\Delta T(q) \, d \LL x 
\cr&
= \Delta Q \int_A F'\big(T^\ast(x+v)\big) -
F'\big(T^\ast(x)\big)\, d \LL
\cr&
= \Delta Q \int_A F'\big(T(q)\pm 2 \epsilon\big) -
F'\big(T(p)\pm 2 \epsilon\big)\, d \LL
\cr&
= \Delta Q \LL(A)\bigg[ F'\big(T(q)\pm 2 \epsilon\big) -
F'\big(T(p)\pm 2 \epsilon\big)\bigg].\cr}$$
Since $F'$ is continuous and $\epsilon$ is arbitrary we infer
$$ F'\big(T(q)\big) - F'\big(T(p)\big)
\le {|v|\over \Delta t^\alpha} = {|p-q|\over \Delta t^\alpha}.$$
The second inequality of conclusion (2) follows from this
inequality together with the estimate given
in appendix A.1.15.  With the continuity of $T$ established
the first inequality of conclusion~(2) follows.  The third
inequality of conclusion~(2) now follows from the first
together with the estimate given in appendix~A.1.16.

We use 2.1, 2.2, 2.3 to infer that
$$\eqalign{
\sup_\O T
&
\le {\hbox{Average}(Q)\over c_{{}_s}} + \hbox{diam}(\O)
\lip(T)
\cr
&
\le {Q_\ast|\O|\over c_{{}_s}} + {n^{1\over 2} \rho_\ast \over 2
c_\ast} {1 \over \Delta t^{\alpha}}\cr}$$
which is the right hand inequality asserted in conclusion~(1).

To check the left hand inequality asserted in conclusion~(1)
we consider $p$ as the center of a  ball of radius
$$r = {\Delta t^\alpha \over 2 C_\ast T(p)^3}$$
and infer from the second inequality of conclusion~(2) that if
$|p-q| \le r$ then
$${1 \over T(p)^3} - {1 \over T(q)^3} \le {C_\ast r \over \Delta
t^\alpha} = {1 \over 2 T(p)^3}, \qquad
{1 \over T(q)^3} \ge {1 \over 2T(p)^3}, \qquad
{1 \over T(q)^2} \ge {1 \over 2^{2\over 3}T(p)^2}.$$
We then use A.1.1 to estimate
$$\eqalign{ {1 \over 2^{2\over 3}T(p)^2} \alpha(n) r^n
& \le 
\int_{{\bf B}^n(p, r)} {1 \over T^2} \, d \LL
\cr & \le
{1 \over c_{{}_s}}\int_{\O \cap \{ x \colon T(x), a_\ast\}}
c_K\, F(T) \, d \LL +
{|\O| \over a_\ast^2}
\cr & \le \EEE\big(\l K \r, \, Q\big) +
{|\O| \over a_\ast^2}
.\cr
}$$
The left hand inequality of conclusion~(1) follows. \E
\bigskip
{\bf 4.3 Theorem (Variations of crystals of $\EEE$ minimizers).}
{\sl
Suppose crystal
$\l K \r \in \KK$ and  heat distribution $Q \in \QQ$
give an $\EEE$ minimizer for some heat distribution
$P$ and time interval $\Delta t$.
Then there is a decreasing
function $\rho \colon \R^+ \to \R^+$ with $\lim_{r \downarrow
0} \rho(r) = 0$ depending only on a lower bound for $\Delta t$
and and upper bound for $\EEE\big(\l K \r, \, Q\big)$ for which
the following is true
Suppose $p \in \O$ and $0 < r << 1$ and
$\l L\r \in \KK$ with $K \tt L \subset \B^n(p, \, r)$.
Then
$$\Phi\big(\partial \l L \r \big)
 -\Phi\big(\partial \l K \r \big)
 - \bigg(\LL(L \sim K) - \LL(K \sim L)\bigg) \, H(T(p))
 \pm  \rho(r) \LL(L \tt K) \ge 0.$$ 
}
\bigskip
PROOF
The assumed minimality of $\l K \r$ and $Q$ implies that
that for every $Q^\ast \in \QQ$,
$$\EEE\big(\l L \r, \, Q^\ast\big)+ \Delta t^{-\alpha} \|Q^\ast
- P\|_\ast
- \EEE\big(\l K \r, \, Q\big)  - \Delta t^{-\alpha}\|Q - P \|_\ast\ge
0.\eqno{(\ast)}$$
Ultimately this inequality $(\ast)$ will give our assertion.

First we estimate
$$\|Q^\ast - P \|_\ast - \| Q - P\|_\ast
\le \| Q^\ast - Q\|_\ast \le 2s \int_\O \left|\Delta Q_1\right|$$

We fix $0 < r < s = 2r << 1$ and set
$$\Delta Q_1(x) = \cases{0 
            & if $x \not\in K \tt L$\cr
\phantom{x}& \cr
{c_{{}_s} - c_{{}_\ell} \over c_{{}_\ell}} Q(x) 
            & if $x \in L \sim K$\cr
\phantom{x}& \cr
{c_{{}_\ell} - c_{{}_s} \over c_{{}_s}} Q(x)
            & if $x \in K \sim L$\cr}$$
and
$$\Delta Q_2(x) = \cases{ 0 & if $x \in \B^n(p, \,r) \cup
\big(\O \sim \B^n(p, \, s)\big)$\cr
\phantom{X} & \cr
\alpha(n)^{-1}\big(s^n - r^n)^{-1} \int_\O \Delta Q_1\, d \LL
&in $x \in \B^n(p,\, s) \sim \B^n(p, \, r)$\cr}$$
and
$$Q^\ast = Q + \Delta Q_1 - \Delta Q_2, \qquad T^\ast = c_L^{-1}
Q^\ast.$$
We check that
$$T^\ast(x) = \cases{T(x)
	& if $ x \in \B^n(p, \, r) \cup \big(\O \sim 
	\B^n(p, \, s)\big)$\cr
\phantom{x}& \cr
T(x) - c_K^{-1}(x) \Delta Q_2(x) & if $x \in \B^n(p, \, s)
\sim \B^n(p, \, r)$\cr}$$

First we estimate
$$\|Q^\ast - P \|_\ast - \| Q - P\|_\ast
\le \| Q^\ast - Q\|_\ast \le 4r \int_\O \left|\Delta Q_1\right|
\le 4r c_{{}_\ell} \sup_\O T \, \LL(K \tt L)$$
which is an admissible error for our  conclusion.

Next we use the Mean Value Theorem to infer the existence of
$T^{0}(x)$ between $T(x) - \Delta Q_2(x)$ and $T(x)$,
for $x \in \B^n(p, \, s) \sim \B^n(p, \, r)$, so that we can
compute
$$\eqalign{\int_\O \big[ c_L F(T^\ast)& - c_KF(T)\big] \, d \LL 
\cr
& = \big( c_{{}_s} - c_{{}_\ell} \big) \int_{L \sim K} F(T) \, d
\LL 
+ \big( c_{{}_\ell} - c_{{}_s} \big) \int_{K \sim L} F(T) \, d
\LL \cr
&\phantom{XXX} +
\int_{{\bf B}^n(p, \, s) \sim {\bf B}^n (p, \, r)}
\left[c_K F\left( T - c_K^{-1} \Delta Q_2 \right) - c_KF(T)\right]
\, d \LL\cr
&
= \big( c_{{}_s} - c_{{}_\ell} \big) \int_{L \sim K} F(T) \, d
\LL
+ \big( c_{{}_\ell} - c_{{}_s} \big) \int_{K \sim L} F(T) \, d
\LL \cr
&\phantom{XXX}-
\int_{{\bf B}^n(p, \, s) \sim {\bf B}^n (p, \, r)}
F'\left( T^{0}\right)  \Delta Q_2 \, d \LL\cr}$$
We can then write
$$
\big( c_{{}_s} - c_{{}_\ell} \big) \int_{L \sim K} F(T) \, d
\LL =
\big( c_{{}_s} - c_{{}_\ell} \big) \LL(L \sim K)
\big[F(T(p)) \pm r\, \lip\big(F\big|\hbox{image}(T)\big)\, \lip(T)\big]$$
$$ \big( c_{{}_\ell} - c_{{}_s} \big) \int_{K \sim L} F(T) \, d
\LL =
\big( c_{{}_\ell} - c_{{}_s} \big) \LL(K \sim L) 
\big[F(T(p)) \pm r\, \lip\big(F\big|\hbox{image}(T)\big)\,
\lip(T)\big]$$
$$ \eqalign{
\int_{{\bf B}^n(p, \, s) \sim {\bf B}^n (p, \, r)}
& F'\left( T^0\right)  \Delta Q_2 \, d \LL
\cr &
=
\left(\int_\O \Delta Q_1\, d \LL \right)\,
\left({1 \over \alpha(n)(s^n - r^n\big)}
\int_{{\bf B}^n(p, \, s) \sim {\bf B}^n (p, \, r)}
F'\left( T^0\right)  \, d \LL\right)
\cr &
=
\bigg[
\big( c_{{}_s} - c_{{}_\ell} \big) \LL(L \sim K)
\big[T(p) \pm 2r\,
%\lip\big(F\big|\hbox{image}(T)\big)\,
\lip(T)\big)\big]
\cr &
\phantom{xx}
+
\big( c_{{}_\ell} - c_{{}_s} \big) \LL(K \sim L)
\big[T(p)
\pm 2r\,
%\lip\big(F\big|\hbox{image}(T)\big)
\lip(T)\big]
\bigg]
\cr &
\phantom{XXXXXXXXXX}
\bigg[F'(T(p)) \pm 2r \, \lip\big(F'\big|\hbox{image}(T)\big)
\lip(T)\bigg]
.\cr}$$
We now incorporate these equalities into the inequality~$(\ast)$.
The theorem follows since the conclusions of Theorem~4.2
give upper and lower bounds for the image of~$T$ and
an upper bound for the Lipschitz constant of~$T$.
\E
\bigskip
{\bf 4.4 Theorem (The Gibbs-Thomson relation for
$\EEE$ minimizers when $\Phi$ is smooth).}
{\sl
Suppose our surface energy integrand $\Phi$ is smooth 
and that $\l K \r \in \KK$ and $Q \in \QQ$ give
an $\EEE$ minimizer for some heat distribution $P$ and some
time  interval $\Delta t$.  Then the Gibbs-Thomson relation
holds with Gibbs-Thomson function~$H$.  This means (as in~2.8)
that for each smooth initial velocity vector field and
associated diffeomorphisms $G_t(x) = x + t g(x)$ we have
$${d \over dt} \Phi\big(G_{t\sharp}\partial \l K \r
\big)\Big|_{t = 0} = \int_{x \in \partial K}
H\big(T(x)\big)\, {\bf n}_K(x) \bullet g(x) \, d \HH x.$$
}
\bigskip
PROOF.  Assume that $\Phi$ is smooth.
As noted in [AW 4.9] the function
$$ g \mapsto {d \over dt} \Phi\big(G_{t\sharp}\partial \l K \r
\big)\Big|_{t = 0}$$
(as above) is well defined and
is a linear functional on the vector space of smooth $g$'s.
In accordance with [AW 4.1(1)] and [FH 4.5.6(5)]
we can compute
$$ {d \over dt} \LL\big(G_t  K\big)\Big|_{t = 0}
= \int_K \div(g) \, d \LL = \int_{\partial K} {\bf n}_K\bullet
g \, d \HH$$
so that the function
$$ g \mapsto {d \over dt} \LL\big(G_t K\big)\Big|_{t = 0}$$
is also well defined 
as a linear functional on the vector space of smooth $g$'s.
We infer by approximation that the function
$$ g \mapsto {d \over dt} \LL\big(G_t( K) \tt K \big)\Big|_{t = 0}$$
is well defined and equals
$$\int_{\partial K} \big| {\bf n}_K \bullet g \big|\, d \HH$$
We now fix an initial velocity vector field $g$.
For a given (small) positive radius $r$ we choose nonnegative functions
$\varphi_1, \, \varphi_2, \, \ldots, \, \varphi_N$
together with points $p_1, \, \ldots, \, p_N$ in $\O$
such that the $\varphi$'s constitute a partition of unity
for $\O$ and the support of each $\varphi_i$ lies in the ball
$\B^n\left(p_i, \, r\right)$.
For each $i$ we set $g_i = \varphi_i\,g$ and corresponding set
$G^i_t(x)= x + t g_i(x)$ for all small $t$'s.
For each fixed $i$ we infer from Theorem~4.3 that for all small $t$'s
$$\Phi\big(G^i_{t\sharp}\partial \l K \r \big)
- \Phi\big(\partial \l K \r \big)
- \bigg(\LL\big( G^i_t(K) \sim K \big) - \LL\big(K \sim G^i_t(K)\big)
\bigg) H\big(T(p_i)\big)$$
$$\phantom{this is a guess on how  far to move}\pm \rho(r) \LL\big( G_t^i(K) \tt K\big) \ge 0.$$
In view of the differentiability established above we further
infer
$$ {d \over dt} \Phi\big(G^i_{t\sharp}\partial \l K \r
\big)\Big|_{t = 0}
- {d \over dt} \LL\big(G_t K \big)\Big|_{t = 0}\,
H\big(T(p_i)\big)
+\rho(r) {d \over dt} \LL\big(G_t(K)\tt K \big)\Big|_{t = 0}
\ge 0$$
so that
$$ {d \over dt} \Phi\big(G^i_{t\sharp}\partial \l K \r
\big)\Big|_{t = 0}
- H\big(T(p_i)\big)
\int_{\partial K} {\bf n}_K \bullet g_i \, d \HH
\pm \rho(r)
\int_{\partial K} \big| {\bf n}_K \bullet g_i \big|\, d \HH\ge
0.$$
We now sum over $i$ to obtain
$$ {d \over dt} \Phi\big(G_{t\sharp}\partial \l K \r
\big)\Big|_{t = 0}
- \sum_{i=1}^N H\big(T(p_i)\big)
\int_{\partial K} {\bf n}_K \bullet g_i \, d \HH
\pm \sum_{i=1}^N \rho(r)
\int_{\partial K} \big| {\bf n}_K \bullet g_i \big|\, d \HH \ge 0$$
so that (since the $\varphi$'s are nonnegative)
$$ {d \over dt} \Phi\big(G_{t\sharp}\partial \l K \r
\big)\Big|_{t = 0}
-
\int_{\partial K} \left(\sum_{i=1}^N
\varphi_i\,H\big(T(p_i)\big)\right)\,
{\bf n}_K \bullet g \, d \HH
\pm \rho(r)
\int_{\partial K} \big| {\bf n}_K \bullet g \big|\, d \HH \ge 0$$
Since our choice of $r$ is arbitrary we use Theorem~4.2 to
infer
$$ {d \over dt} \Phi\big(G_{t\sharp}\partial \l K \r
\big)\Big|_{t = 0}
- \int_{\partial K} H(T)\, {\bf n}_K \bullet g \, d \HH \ge 0.$$
Since a similar inequality must hold with $g$ replaced by $-g$
we infer
$$ {d \over dt} \Phi\big(G_{t\sharp}\partial \l K \r
\big)\Big|_{t = 0}
- \int_{\partial K} H(T)\, {\bf n}_K \bullet g \, d \HH = 0$$
which gives the claim of the theorem.  \E
\bigskip
{\bf 4.5 Theorem (Regularity of $\EEE$ minimizers).}
{\sl
Suppose that $\l K \r \in \KK$ and $Q \in \QQ$ give
an $\EEE$ minimizer for some heat distribution $P$ and some
time interval $\Delta t$.  Suppose also that $T$ is the
temperature field associated with $K$ and $Q$.
Then
\smallskip\noindent
(1) There exist positive numbers $\delta$ and $\mu$ such that
$$\HH \big(\partial K \cap \B^n(p, \, r)\big) \ge \mu\,
r^{n-1}$$
for each point $p$ in the support of $\partial \l K \r$ and 
each radius $0 < r < \delta$.
\smallskip\noindent
(2) The support of $\partial \l K \r$ (which equals the
closure of $\partial K$) has finite $\HH$ measure and hence
zero $\LL$ measure.
\smallskip\noindent
(3) There exist positive numbers $\delta$ and $C$ together
with a function $\omega (r) = C\, r$ defined for $0 < r <
\delta$ such that the current $\partial \l K \r$ is
$\big( \Phi, \, \omega, \, \delta \big)$-minimal in  the sense
of E.~Bombieri [BE Definition~1, page 101].
\smallskip\noindent
(4) In case $\Phi$ is an even integrand there exist positive
numbers $\gamma$ and $\delta$ such that the support of
$\partial \l K \r$ is $(\gamma, \, \delta )$ restricted with
respect to the empty set in
the sense of F.~Almgren [AF II.1, page~53]
\smallskip\noindent
(5) In case $\Phi$ is an even integrand there exist positive
numbers $\delta$ and $C$ together with a function $\epsilon(r)
= C\, r$ defined for $0 < r < \delta$ such that the support
of $\partial \l K \r$ is $(\Phi, \, \epsilon, \, \delta)$
minimal with respect to the empty set in the sense of
F.~Almgren [AF III.1, page~75]
\smallskip\noindent
Now suppose additionally that our surface energy integrand $\Phi$
is smooth and elliptic.  Then as additional conclusion we
have that
\smallskip\noindent
(6) Except possibly for a compact singular set of zero $\HH$
measure, the support of $\partial \l K \r$ is a two times
H\"older continuously $n-1$~ dimensional submanifold of~$\O$.
\smallskip\noindent
(7) If $n=2$ or $n = 3$ then the support $S$ of $\partial \l K \r$
is a two times H\"older continuously differentiable
submanifold of $\O$ (with no singular set).  Furthermore,
at every point $p$ of $S$ the weighted mean curvature $wmc(p)$
of $S$ (with respect
to the exterior normal of $K$) exists in the classical sense
and equals $H(T(p))$. 
}
\bigskip
PROOF.  The proof of the present theorem very closely follows the 
arguments and constructions of sections 3.5, 3.5, 3.6, 3.8,
3,9, 3,10 of the paper [ATW] mentioned in the introduction in
which a similar variation scheme was used by the present
authors together with  J.~Taylor to produce curvature
driven motions.  Since there are essentially no new ideas
required in adapting the arguments there to the present case,
we leave verification of the present  theorem to the reader.
We note, however, that the first conclusion together with
[FH~2.10.19(1)] imply the second conclusion.
\E
\bigskip
\centerline{\larger 5. Heat flows}
\bigskip
In this chapter we study the existence and regularity of heat
flows with fixed crystals.  Suppose $\l K \r$ and $Q$ give an
$\EEE$ minimizer for some $P$ and $\Delta t$ in the
construction of approximating evolutions in 2.11.
According to conclusion~(2) of Theorem~4.5 the closure of
the boundary $\partial K$ will have zero $n$~dimensional
measure.  Our first theorem below shows that this condition
is sufficient to guarantee the existence of a heat flow in the
sense of~2.5.  
Our second theorem gives integral inequalities
for such flows with fixed crystals.
In our third theorem we establish the essential
H\"older continuity in the Monge-Kantorovich metric
of the heat flows which arise in our  approximating evolutions.
According to conclusion~(7) of Theorem~4.5 the boundary of $K$
will be a H\"older continuously differentiable
submanifold of $\O$ in case our ambient manifold $\O$ is of
dimension either two  or three and our surface energy function $\Phi$
is smooth and elliptic.  Under this regularity assumption  on
$\partial K$ we show in our fourth theorem below that the
heat flow within the crystal and without is H\"older
continuously differentiable in space and H\"older continuous
in time; in particular, there is  temperature continuity with
simple gradient jumps across the interface..
\bigskip
{\bf 5.1 Theorem (Existence of heat flows with fixed
crystal).}
{\sl
Suppose that $\l K \r$ is a crystal in $\KK$ for which the closure 
of $\partial K$
has zero $\LL$ measure.  Suppose also that $a \in \R$ and that $Q_a$ 
is a heat 
distribution in $\QQ$
such that the associated temperature field $T_a = c_{{}_K} Q_a$ 
belongs to
$\H^1(\O)$.  Then there are unique functions
$Q(x, \, t)$ and $T(x, \, t)$ defined for $x$ in $\O$ and
$a \le t < \infty$ such that for each fixed number $b > a$ the 
restriction of
$Q$ and $T$ to $\O \times [ a, \, b]$ are a solution to the heat 
equation
with fixed crystal $\l K \r$ and with initial heat distribution $Q_a$ 
and final
heat distribution $Q(\cdot, \, b)$ in the sense of 2.5.
Furthermore, the functions $Q$ and $T$ are infinitely differentiable 
on the set
$$\bigg(\O \sim \hbox{\rm Closure}(\partial K)\bigg) 
\times (a, \, \infty).$$
Finally, the equation~2.5.7 
$$0 = \int_{\O \times (a, \, b)} -\dotf Q +
\kappa_{{}_K}(\nabla \varphi)\bullet \nabla T\, d \LLL
+ \int_\O \varphi(\cdot, \, b) Q_b - \varphi(\cdot, \, a) Q_a \, d \LL$$
holds whenever
$$\varphi \in {\bf C}^0\big([a,\, b], \, \L_2(\O)\big) \cap
\L_2\big( (a, \, b) , \, \H^1(\O)\big)\quad
\hbox{with}\quad\dotf \in \L_2\big([a, \, b], \, \L_2(\O)\big).$$
}
\bigskip
PROOF.  Our existence proof ultimately is based on the theory of
analytic semigroups as set forth in [PA].  There are five
steps.
\smallskip
{\bf Step 1.}
%The spaces $\LK$ and $\HK$ and operator $A$.
We denote by $\LK$ the vector space $\L_2(\O)$
with  inner product
$\big(\cdot\,, \, \cdot)_K$ and norm $\|\cdot\|_{K}$
given by setting for $u, \, v \in \L_2(\O)$,
$$\big(u, \, v\big)_K = 
\int_{x \in \O} c_K(x)\, u(x)\, v(x)\, d \LL x, \quad \|u\|_K^2
= \big(u, \, u\big)_K;$$
here $c_K$ is our heat capacity function which equals
$c_{{}_s}$ in $K$ and equals $c_{{}_\ell}$ in $\O \sim K$.
\smallskip
We define a linear partial differential operator
$$Au = -{1 \over c_K} \div\big(\kappa_K\, \nabla u\big)$$
taking values in $\LK$ and
having as domain in $\LK$ the vector space,
$$\D(A) = \H^1(\O) \cap \big\{ u \colon Au \in \LK\big\}\subset
\LK.$$
When we write $Au \in \LK$ we mean that there exists
a function $f$ in $\LK$, which symbolically is denoted $Au$,
such that
$$\big(Au, \, v\big)_K = \big(f, \, v\big)_K
= \int_{x \in \O} \big(\kappa_K(x) \nabla u(x)\big) \bullet
\nabla v(x)\, d \LL x$$
for each $v \in \H^1(\O)$.
Here $\kappa_K$ is heat conductivity which equals
the symmetric matrix
$\kappa_{{}_s}$ in $K$ and equals
the symmetric matrix $\kappa_{{}_\ell}$ (a multiple of the identity)
in $\O \sim K$.
\smallskip
Additionally we define a Hilbert space
$$\HK$$
which as the vector space consists of those functions $u$ in
$\H^1(\O)$ for which
$$\int_{x \in\O} u(x)\, c_K(x)\, d \LL x = 0$$
and which carries the inner product $\big<\cdot\,, \, \cdot\big>_K$
given by setting, for each $u$ and $v$,
$$\big< u, \, v\big>_K = \int_{x \in \O}\big(\kappa_K(x)\,
\nabla u(x)\big) \bullet \nabla v(x)\, d \LL x.$$
$\HK$ is a Hilbert space since Poincar\'e's inequality guarantees that
$$\int_\O u^2\, c_K\, d \LL \le \Gamma^2 
 \int_\O\big(\kappa_K\, \nabla u\big) \bullet \nabla v\, d \LL$$
\bigskip
{\bf Step 2.}  We assert that $\D(A)$ is dense in $\LK$.  
Since $\DD\big(\O\big)$ is dense in $\LK$ it is sufficient to show that
$\DD\big(\O\big)$ lies in the  closure of $\D(A)$.  For small
positive $\epsilon$'s we pick smooth functions
$\varphi_\epsilon\colon \O \to [0, \, 1]$
such that $\varphi_\epsilon^{-1}\{1\}$ contains a
neighborhood of the closure of $\partial K$ and $\int_\O
\varphi_\epsilon \, d \LL < \epsilon.$
Suppose then $g \in \DD\big(\O\big)$.  Clearly
$g \big( 1 - \varphi_\epsilon\big) \in \D(\O)$ and
$\lim_{\epsilon\downarrow 0} \|g - g(1 - \varphi_\epsilon)\|_K =0$.
\bigskip
{\bf Step 3.}  We assert that for each function $w_0$ in $\LK$
with
$$\int_\O c_K \, w_0 \, d \LL = 0$$
the equation $Au = w_0$ has a unique solution $u = u_0$ in
$\D(A)$.
\medskip
{\bf Proof.} The function
$$ v \mapsto \big(v, \, w_0\big)_K = \int_\O v\, w_0\, c_K \,
d \LL$$
is a continuous linear functional on $\HK$ since
$$\bigg|\big(v, \, w_0\big)_K\bigg|
\le \left({\int_\O v^2 \, c_K \, d \LL}\right)^{1 \over 2}
\left({\int_\O w_0^2 \, c_K \, d \LL}\right)^{1 \over 2}
\le \Big(\Gamma\, \|w_0\|_K\Big)
\big< u, \, u\big>_K^{1 \over 2}$$
by Poincar\'e's inequality above.  The usual application of 
Riesz's Representation Theorem guarantees the existence
of $u_0$ in $\HK$ such that, for each $v$ in $\HK$,
$$ \big(v, \, w_0\big)_K = \big<v, \, u_0\big>_K.$$
If $v \in \H^1(\O)$ and $\gamma = \int_\O v \, c_K \, d\LL/
\int_\O c_K\, d \LL$,
then $v - \gamma \in \HK$.  Since
$$ \big(\gamma, \, w_0\big)_K = \big<\gamma, \, u_0\big>_K = 0$$
whenever $\gamma$ is constant we infer that $Au_0 = w_0$.
\bigskip
{\bf Step 4.} We assert that $A$ is a positive operator, i.e.
that $A$ is both positive  and self-adjoint.
\medskip
{\bf Proof.}  It is clear from our definition above that
$A$ is positive, i.e. $\big(Au, \, u\big)_K \ge 0$ for
each~$u$.  It is also clear that $A$ is symmetric since
$$\big(Au, \, v\big)_K = \int_\O \nabla u \, \kappa_K \,
\nabla v \, d \LL = \big(u, \, Av\big)_K.$$
Suppose then $A^\ast$ is the adjoint of $A$ and that
$\D(A^\ast)$ is the domain of $A^\ast$ so that
$$\big(Au, \, v\big)_K = \big(u, \, A^\ast v\big)_K\eqno{(\ast)}$$
for each $u \in \D(A)$ and each $v \in \D(A^\ast)$.  Since
$A$ is symmetric, $\D(A) \subset \D(A^\ast)$.  To show that
$A= A^\ast$ we must show that $v_0$ in $\D(A^\ast)$
implies $v_0 \in \D(A)$.
Since the  constant function $u = 1$ belongs to $\D(A)$ we
infer from $(\ast)$ that
$$0 = \big(A\,1, \,v_0 \big)_K
= \big(1, \, A^\ast v_0\big)_K = \int_\O A^\ast v_0 \,c_K \, d \LL$$
We infer from Step~3 that there is a unique $u_0$ in $\HK$
such that 
$$Au_0 = A^\ast v_0;\eqno{(\ast)}$$
clearly $u_0 \in \D(A)$ with $\int_\O u c_K\, d \LL =0$.
For each $u \in \D(A)$ we use $(\ast)$ to check
$$\big(Au, \, v_0 - u_0 \big)_K
=\big(Au, \, v_0 \big)_K
-\big(Au, \, v_0 \big)_K
=\big(u, \, A^\ast v_0 \big)_K
-\big(u, \, Au_0 \big)_K
=0\eqno{(\ast\ast)}$$
We now set
$$\gamma = {\int_\O v_0 \, c_K \, d \LL \over \int_\O c_K \, d
\LL}$$
and
$$w_0 = v_0 - u_0 - \gamma$$
so that Step~3 again guarantees the existence of a unique
$z_0$ in $\HK$ such that $Az_0 = w_0$.  Hence we use
$(\ast\ast)$ to infer
$$0 = \big(Az_0, \, v_0 - u_0 \big)_K
= \big( v_0 - u_0 - \gamma, \, v_0 - u_0 \big)_K
= \big( v_0 - u_0 - \gamma, \, v_0 - u_0 - \gamma \big)_K$$
which implies that
$$ v_0 = u_0 + \gamma$$
so that $v_0 \in \D(A)$ with
$$Av_0 = Au_0 = A^\ast v_0$$
in accordance with $(\ast)$.
\bigskip
{\bf Step 5.} We assert that the operator $-A$ is the
generator of a $C_0$ {\bf semigroup} in the sense of Definition~2.1
on page 4 of [PA].  This is a consequence of Theorem~3.1
(Hille-Yosida) on page 8 of [PA] once we establish that

(i) $-A$ is closed with the closure of $\D(-A)$ equal to $\LK$.

(ii) the resolvent set $\rho(-A)$ contains $\R^+$,

(iii) for every $\lambda >0$ the mapping norm of $(\lambda I +
A)^{-1}$ does not exceed $\lambda^{-1}$.

First $-A$ is closed because $A= A^\ast$ and the adjoint
operator is always closed, and we have already established
that $\D(A)$ is dense in $\LK$.

We recall that $\rho(-A)$ equals the set of $\lambda$'s for
which the operator $\lambda I + A$ is invertible.  Since $A$
is positive, we estimate for $x$ in $\D(A)$,
$$\big((\lambda I + A)x, \, x\big)_K
= \lambda \big(x, \, x\big)_K + \big(Ax, \, x\big)_K
\ge \lambda \big(x, \, x\big)_K$$
and infer that for all positive $\lambda$, $(\lambda I + A)$
is invertible and the inverse  satisfies the estimate in (iii).
\bigskip
{\bf Step 6.}  We assert that the operator $-A$ is the
generator of an {\bf analytic semigroup} in the sense of
Definition~5.1 on page 60 of [PA].  This is a consequence of
the condition (b) of Theorem~5.2 on page 61 of [PA].  In a
manner similar to our estimate  in Step 5 we estimate
$$\left|\bigg(\big((\sigma + {\bf i}\tau)I + A\big)x,\,x\bigg)_K\right|
= \left|\sigma \big(x, \, x\big)_K
+ {\bf i} \tau \big(x, \, x\big)_K
+ \big(Ax, \, x\big)_K \right|
\ge\left| \tau\right| \big(x, \, x\big)_K$$
so that the mapping norm of the inverse to
$$\bigg(\big((\sigma + {\bf i} \tau) I + A\big)x, \, x\bigg)_K$$
does not  exceed $|\tau|$ as required.
\bigskip
{\bf Step 7.}  We assert that the evolution equation
$$c_K\,\dott = \div\big( \kappa_K \, \nabla T\big)$$
$$T(x, \, 0) = T_0(x)$$
is uniquely solvable for any initial conditions $T_0$
belonging to $\LK$.  This is  a consequence of Corollary~1.5
on page~104 of [PA].
\bigskip
{\bf Step 8.} We assert that $-A$ is the generator of a {\bf
classical semigroup} in the sense of Definition~2.1 on page~104
of [PA].  This is a consequence of Theorem~3.2 on page~111
of [PA].
\bigskip
{\bf Step 9}  We infer from Step~8 that for $0 < a < b <
\infty$,
$$\int_\O T^2(\cdot, \, b)c_K \, d \LL
-\int_\O T^2(\cdot, \, a)c_K \, d \LL
=- \int_{\O\times(a, \, b)} \nabla T \kappa_K \nabla T\, d \LLL.$$
We use Lebesgue's increasing convergence theorem to infer that
our equality also holds for $a = 0$.
Hence, 
$$\int_{\O\times(0, \, b)} \nabla T \kappa_K \nabla T\, d \LLL< \infty$$
for each $b$.
We assert that for each $0 \le a < b <
\infty$ and each $\varphi \in \DD(\O\times \R)$
$$\int_{x \in \O} \varphi(x, \, b)\, c_K(x)\, T(x, \, b)\, d \LL
+ \int_{(x, \, t) \in \O \times [a, \, b]} 
\nabla \varphi(x, \, t)\, \kappa_K(x)\, \nabla T(x, \, t)
\, d\LLL(x, \, t)$$
$$\phantom{X}
= \int_{x \in \O} \varphi(x, \, a)\, c_K(x)\, T(x, \, a)\, d \LL
+\int_{(x, \, t) \in \O \times [a, \, b]} 
\dotf(x, \, t)\, c_K(x)\, T(x, \, t) \, d\LLL(x, \, t).$$ 
For $a > 0$, our assertion is essentially an identity.
Clearly it holds for $a = 0$ as well since our semigroup is
continuous.  The final assertion of our theorem above follows
by approximation.
\bigskip
{\bf Step 10.}  The regularity asserted for our solution is a
consequence of Theorem~10.1 on page 351 of [LSU].
\E
\bigskip
{\bf 5.2 Theorem (Inequalities under heat flow).}
{\sl Suppose that $Q(x, \, t)$ and $T(x, \, t)$ defined on $\O \times
[a, \, b]$ are a solution to the heat equation with fixed crystal $\l K \r$
and with initial heat distribution $Q_a$ and final heat distribution
$Q_b$ as in 2.5.  Also set $U = 1/T$.
\medskip\noindent
(1) For each twice continuously differentiable function convex function
$G \colon \R^+ \to \R$ we have
$$\int_\O c_{{}_K} G\big(c_{{}_K}^{-1} Q_a\big) \, d \LL
 - \int_\O c_{{}_K} G\big(c_{{}_K}^{-1} Q_b\big) \, d \LL
= \int_{\O \times [a, \, b]} G''(T) \, \big(\nabla T \kappa_{{}_K} \nabla T\big) \, d \LLL$$
provided the first term on the left hand side is finite.
\medskip\noindent
(2) For the bulk energy density function $F$ we have
$$\eqalign{\int_\O c_{{}_K} F\big(c_{{}_K}^{-1} Q_a\big) \, d \LL
& - \int_\O c_{{}_K} F\big(c_{{}_K}^{-1} Q_b\big) \, d \LL
\cr &\ge \kappa_\ast\int_{\O \times [a, \, b]} \big(c_\ast + C_\ast^{-1}T^{-4}\big)
 \, |\nabla T|^2 \, d \LLL
\cr & = \kappa_\ast\int_{\O \times [a, \, b]} \bigg(c_\ast  |\nabla T|^2
+ C_\ast^{-1} |\nabla S|^2\bigg) \, d \LLL.\cr }
.$$
\medskip\noindent
(3) For each Lipschitz function $\varphi \colon \O \to \R$,
$$\eqalign{\int_\O \varphi\big( Q_a - Q_ b\big)\, d \LL
&\le \kappa_0 \lip(\varphi)
\int_{\O \times [a, \, b]} |\nabla T| \, d \LLL
\cr & \le \kappa_0 \lip(\varphi)|\O|^{1 \over 2} (b - a)^{1 \over 2} \left(
\int_{\O \times [a, \, b]} |\nabla T|^2 \, d \LLL\right)^{1 \over 2}.\cr}$$
\medskip\noindent
(4) As another estimate on changes in temperature under heat flow we have
$$\eqalign{
\int_\O \left| \kappa_{{}_K}^{1 \over 2} \nabla T(\cdot, \, a) \right|^2 \, d \LL
-\int_\O \left| \kappa_{{}_K}^{1 \over 2} \nabla T(\cdot, \, b) \right|^2& \, d \LL
\cr & = 2 \int_{(x, \, t) \in\O \times [a, \, b]} c_K(x) \dott(x, \, t)^2 \, d \LLL(x, \, t).
\cr }$$
\medskip\noindent
(5) As a consequence of (4) we have
$$\int_\O \big(T(\cdot,\, a) - T(\cdot, \, b)\big)^2 \, d \LL
\le {\kappa_0 \over 2 c_{{}_s}} (b - a) \, \dir\big( T(\cdot, \, a) \big).$$
}
\bigskip
PROOF. With regard to assertion (1)
we first suppose the second derivatives of
$G$ are bounded.  In that case we use the  final equation of
Theorem~5.1 with $\varphi = G'(T)$ together with 
integration by parts to infer
$$\int_\O G(T(\cdot, \, b))\, c_K \, d\LL + \int_{\O\times (a, \,
b)}\G''(T)\, \nabla T \, \kappa_K \nabla T \, d \LLL =
\int_\O G(T(\cdot, \, a))\, c_K \, d\LL.$$
The assertion of (1) now  follows by approximation.
Note also the calculation
$$\eqalign{-{d \over dt} \int_\O c_{{}_K} G\big(c_{{}_K}^{-1} Q(\cdot, t) \big) \, d \LL
&= -\int_\O c_{{}_K} G'\big(c_{{}_K}^{-1}Q(\cdot, \, t)\big) c_{{}_K}^{-1}\dotq(\cdot, \, t)
\, d \LL
\cr &= - \int_\O G'\big(T(\cdot, \, t)\big) \div\big(\kappa_{{}_K}\nabla T(\cdot, \, t)
\big) \, d \LL
\cr &= \int_\O G''\big(T(\cdot, \, t)\big)\, \big(\nabla T(\cdot, \, t) \kappa_{{}_K}
\nabla T(\cdot, \, t)\big) \, d \LL.\cr}$$
Assertion (2) follows from assertion (1) and inequality A.1.10.

With regard to assertion (3) we compute
$$\eqalign{\int_\O \varphi\big( Q_a - Q_b\big)\, d \LL
&=- \int_{\O\times [a, \, b]} \varphi\big(\dotq\big)\, d \LLL\cr
&=- \int_{\O\times [a, \, b]} \varphi\, \div\big(\kappa_{{}_K} \nabla T \big)\, d \LLL\cr
&= \int_{\O\times [a, \, b]} \nabla \varphi\kappa_{{}_K} \nabla T \, d \LLL\cr
&\le \kappa_0 \lip(\varphi) 
 \int_{\O\times [a, \, b]} |\nabla T| \, d \LLL\cr}$$
and then use Schwarz's inequality. 

With regard to assertion (4) we compute
$$\eqalign{\int_\O c_{{}_K} \dott^2 \, d \LL 
& = \int_\O \dott \, \div\big(\kappa_{{}_L} \nabla T\big)
\, d \LL 
\cr & = - \int_\O \nabla \dott \kappa_{{}_K} \nabla T \, \LL
\cr &= - {1 \over 2} \int_\O {\partial \over \partial t}\bigg(\big| \kappa_{{}_K}^{1 \over 2}
\nabla T \big| \bigg)\, d \LL.\cr}$$

With regard to assertion (5) we compute for $\LL$ almost
every $x$ in $\O$,
$$\left| T(x, \, b) - T(x, \, a) \right|
\le \left|\int_{ t \in [a, \, b]} \dott(x, \, t) \, d {\cal L}^1 t\right|
\le (b - a)^{1 \over 2}\left( \int_{ t \in [a, \, b]} \dott(x, \, t)^2 \, d {\cal L}^1 t
\right)^{1 \over 2}$$ 
and use assertion (4) to estimate
$$(b-a)^{-1} \int_{ x \in \O}
\left| T(x, \, b) - T(x, \, a) \right|^2 \, d \LL x
\le \int_{\O \times [a, \, b]} \dott^2 \, d \LLL
\le {\kappa_0\over 2 c_{{}_s}} \int_\O \left| \nabla T(\cdot, \, a)\right|^2 \, d \LL.
$$ \E
\bigskip
{\bf 5.3 Theorem (H\"older estimates for heat flows of approximating evolutions).}
{\sl
Suppose $j$ is a positive integer, $\Delta t = 2^{-j},$  and $0 < s < t < \infty$.  
Then
\smallskip\noindent
(1) $\EE_j(s) \ge \EE_j(t)$
\smallskip\noindent
and
\smallskip\noindent
(2) In our approximating heat evolution $Q_j(x, \, t)$, we have
$$\big\|Q_j(\cdot, \, t) - Q_j(\cdot, \, s) \big\|_\ast \le
\EE_0\, \Delta t^\alpha +
 {\kappa_0 |\O|^{1 \over 2} 
\over \kappa_\ast^{1 \over 2} c_\ast^{1 \over 2}}
 (t-s)^{1 \over 2}\,\EE_0^{1 \over 2}.$$
}
\bigskip
PROOF.
Suppose $k$ is a positive integer and $t_0 = k \Delta t$.  
In construction of our approximating evolutions we always have the
option of taking $\l K_j(t_0) \r = \l K_j(t_0 - \Delta t) \r$ and
$Q_j(t_0) = P_j(t_0)$.  Hence, it will always be true that
$$\EE_j\big(t_0^-\big) - \EE_j\big(t_0\big) \ge {1 \over \Delta t^\alpha}
\|Q_j(t_0) - P_j(t_0)\|_\ast \ge 0.$$
Such inequalities together with conclusion (2) of Theorem~5.2
imply the first assertion above.
Furthermore, summation of such inequalities for $t_0$'s between $s$ and $t$.
is dominated by the
first term on the right hand side of the inequality asserted in (2) above.

Now suppose $t_0 - \Delta t \le r < t_0$.  We use conclusions (3) and (2)
of Theorem~5.2 to infer
$$\eqalign{\big\|Q_j(\cdot, \, r) - P_j(\cdot, \,t_0) \big\|_\ast
&\le \kappa_0 |\O|^{1 \over 2} (t_0 - r)^{1 \over 2}
\left(\int_{\O \times [r, \, t_0]}
|\nabla T_j|^2\, d \LLL \right)^{1 \over 2}
\cr & \le
 {\kappa_0 |\O|^{1 \over 2}
\over \kappa_\ast^{1 \over 2} c_\ast^{1 \over 2}}
 \big(t_0 - r\big)^{1 \over 2} 
\left( \EE_j(r) - \EE_j(t_0^-)\right)^{1 \over 2}.\cr}$$
A corresponding inequality holds if $t_0 < r \le t_0 + \Delta t$.
The second term on the right hand side of the inequality asserted in (2) above
comes from
summation of terms associated with intervals $\left[t_0, \, t_0 + \Delta t\right]$,
etc. for $t_0$'s between $s$ and $t$ and a straight forward application of
Schwarz's inequality for sums, viz
$$\sum_i\Delta t_i^{1 \over 2} \Delta E_i^{1 \over 2} \le 
\left(\sum_i \Delta t_i\right)^{1\over 2}\left( \sum_i \Delta E_i \right)^{1 \over 2}$$
for positive numbers $\Delta t_i$ and $\Delta E_i$. \E
\bigskip
The terminology of the following theorem is that common in the
theory of linear parabolic partial differential equations.
\bigskip
{\bf 5.4 Theorem (Regularity of heat flow across an interface).}
{\sl  Suppose
\smallskip\noindent
(a) $\T = \U^{n-1}(0, \, 2) \times \U^1(0, \, 2)$
\smallskip\noindent
(b) $0<\alpha < 1$ and $\sigma\colon\, \U^{n-1}(0, \, 2) \to
\U^{1}(0, \, 1/2)$ is H\"older continuously differentiable with
exponent $\alpha$.
%with $\sigma(0, \, \ldots, \, 0) = 0$ 
%and
%$\nabla \sigma(0, \, \ldots, \, 0) = 0$. 
The means that
$$\Gamma = \T \cap \big\{ x \colon x_n
= \sigma\left(x_1, \, \ldots, \, x_{n-1}\right)\big\}$$
is a H\"older continuously differentiable hypersurface in
$\T$ and
$$\T^+ = \T \cap \big\{ x
\colon x_n > \sigma\left(x_1, \, \ldots, \, x_{n-1}\right)\big\},
\quad
\T^- = \T \cap \big\{ x
\colon x_n < \sigma\left(x_1, \, \ldots, \, x_{n-1}\right)\big\}$$
are regions which share the boundary $\Gamma$. 
\smallskip\noindent
(c) $\gamma^0 \colon \T \sim \Gamma \to [1, \, \infty)$ is
constant in each component of $\T \sim \Gamma$;
we also write $\gamma^0(x, \, t) = \gamma^0(x)$.
\smallskip\noindent
(d) $\kappa^0\colon \T \sim \Gamma \to \hbox{\rm Hom}\left(\R^n
, \, \R^n\right)$ is constant in each component of $\T \sim \Gamma$.
For each~$x$, $\kappa^0(x) =
\left\{\kappa^0_{ij}(x)\right\}$ is a symmetric $n\times n$ matrix
with $v \bullet \big<v, \, \kappa^0(x)\big> \ge |v|^2$ for each~$v$; we
also write $\kappa^0(x, \, t) = \kappa^0(x)$.
\smallskip\noindent
(e) $u \colon \T \times (-2, \, 0] \to \R$ is a strictly
defined function which is a weak solution
of the partial differential equation
$$\gamma^0(x){\partial u \over \partial t}(x, \, t)
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa^0_{ij}(x)\, {\partial u\over \partial
x_i}(x, \, t) \right). $$
By this we mean that $u$ belongs to $\C^0\big((-2, \, 0], \,
\H^1(\T)\big)$ and that, for each test function $\varphi$ in $\DD\big(
\T \times (-2, \, 2)\big)$ and each $-2 < a < b \le 0$,
$$\eqalign{0 = \int_{(x, \, t) \in \T \times (a, \, b)}
\Bigg(-\gamma^0(x)\,{\partial \varphi \over \partial t}&(x,\, t)
\,\,u(x,\,t)
\cr 
+\sum_{j=1}^n{\partial \varphi\over \partial x_j}(x, \, t)
&
\sum_{i=1}^n \left(\kappa^0_{ij}(x)\,{\partial u\over\partial x_i}(x, \, t)
\right)\Bigg)  \, d {\cal L}^{n+1}(x, \, t)
\cr &
+ \int_{x \in \T} \gamma^0(x)\big(\varphi (x, \, b)\, u(x, \, b)
-\varphi(x,\,a)\, u(x, \, a)\big)\, d \LL x.\cr}$$
Then $u$ is continuous and,
corresponding to each compact subset $K$ of $\T \times
(-2, \, 0]$, there is a constant $C_K$ with the following
property:  Suppose
$(x, \, s)$ and $(y, \,s)$ and $(y,\, t)$ belong to $K$
such that either $x$ and $y$ both belong to $\T^+$ or they
both belong to $\T^-$.  Then
$$\|\nabla u(x, \, s) - \nabla u(y, \, s) \| \le C_K |x -
y|^\alpha \quad\hbox{and}\quad |u(y, \, s) - u(y, \, t)| \le
C_K \, |s - t|^{1 + \alpha\over 2};$$
$$\hbox{and}\quad \|\nabla u(y, \, s) - \nabla u(y, \, t) \|
\le C_K |s - t|^{\alpha\over 2}.$$
In particular, the restriction of $u$ to $\T^+ \times (-2,\, 0] $
$\big[$resp. to $\T^- \times (-2, \, 0]$ $\big]$ has
an extension to a neighborhood of $\T^+ \cup \Gamma\times (-2,
\, 0]$
$\big[$resp. of $\T^- \cup \Gamma\times (-2, \, 0]$ $\big]$
for which the inequalities above
still hold.
}
\bigskip
PROOF.
For $0 < r \le 1$ and $(x, \, t) \in \R^n \times \R$ we set
$$Q_r = \B^n(0, \, r) \times (-r^2,\, 0]\quad\hbox{and}\quad
Q_r(x, \, t) = Q_r + \{(x, \, t)\} \,\, \hbox{(set addition)}.$$
In order to have notation for the average value of a function
we set
$$\aint_{A} h\, d \LLL =
{1 \over \LLL(A)} \int_A h\, d \LLL$$
for appropriate $h$ and $A$.
Our proof is in several steps.  We will use $C$ to denote a
general constant which depends only on $n$ and bounds for
$\sigma$, $\gamma^0$ and $\kappa^0$ and choices of cutoff functions,
etc.  whose value may be
different in different places, even in the same formula.
\medskip
{\bf Step 1 (Application of Campanato's embedding theorem).}
In Step~5 below we introduce a H\"older continuously
differentiable change of coordinates $\psi$  for our domain $\T$ and
hence for our 
function $u$.  In these new coordinates our interface $\Gamma$
becomes part of the hyperplane $x_n = 0$, and
our function $u$ becomes a new function $U(x, \, t) =
u\big(\psi^{-1}(x), \, t\big)$ 
which satisfies a new perturbed partial
differential  equation.  We denote by $\Xi^+$ the domain 
$\psi(\T^+) \cap \B^n(0, \, 1) \times (-1, \,0]$ and by
$\Xi^-$ the domain
$\psi(\T^-) \cap \B^n(0, \, 1) \times (-1, \,0]$, and by 
$\Xi$ the domain
$\psi(\T) \cap \B^n(0, \, 1) \times (-1, \,0]$; the domains $\Xi^\pm$
are  Lipschitz domains.  According 
to S. Campanato's Theorem~2.I in [CS] we know the following:

Suppose $U \colon  \Xi\to \R$ is square summable and strictly
defined and additionally has square summable first
derivatives.  Suppose also
$$\aint_{Q_r(x, \, t)\cap \Xi^\pm} \left|\nabla U - \overline{\nabla
U}\right|^2 \, d \LLL \le C r^{2 \alpha}\eqno{(\ast)}$$
for each $(x, \, t)$ in $\Xi^\pm\cup \partial \Xi^\pm$ and
each $0 < r \le 1$.  Here and elsewhere we denote
by $\overline{\nabla U}$ the average value of $\nabla U$ over
the domain under consideration.

Then there is a constant $C_0$ such
that that whenever
$(x, \, s)$ and $(y, \,s)$ and $(y,\, t)$ belong to $\Xi^\pm$, then
$$\|\nabla U(x, \, s) - \nabla U(y, \, s) \| \le C_0 |x -
y|^\alpha \quad\hbox{and}\quad |U(y, \, s) - U(y, \, t)| \le
C_0 \, |s - t|^{1 + \alpha\over 2}.$$
$$\hbox{and}\quad \|\nabla U(y, \, s) - \nabla U(y, \, t) \|
\le C_0 |s - t|^{\alpha\over 2}.$$

This conclusion together with assertion~(2) of Step~3 below
imply the assertions of the present
theorem.  We will show below that the condition $(\ast)$ is implied
by the conditions (a), (b), (c), (d) below for some finite
constant $C$.  It is these four conditions which we will then
prove.  In order to state
these conditions we define a parabolic distance
function $\delta_\star$ by setting
$$\delta_\star \big((x, \ s)\, , \, (y,\, t) \big)
=\cases{ \big( |x - y|^2  + |s - t|\big)^{1 \over 2}& if $s
\ge t$\cr
\phantom{x} &\cr
\infty & if $s < t$\cr}.$$
Our conditions are the following:
\smallskip\noindent
(a) Corresponding  to each $(x_0, \, t_0) \in \Xi^\pm$ with 
$$s = \delta_\star\big((x_0, \, t_0), \, \partial \Xi^\pm \big)
= \inf\bigg\{ \delta_\star \big((x_0, \, t_0), \, (x, \,
t)\big)\colon (x, \, t) \in \partial \Xi^\pm \bigg\} <
\infty$$
there is a polynomial function $L^\pm_0 \colon \R^n \to \R$ of degree
one with
$$|L^\pm_0(x_0)| + \|\nabla L^\pm_0(x_0)\|\, s
\le C \left(\aint_{Q_{s/ 2}(x_0, \, t_0)} U^2 \, d
\LLL\right)^{1 \over 2}$$
such that for each $0 < r \le s$
$$\aint_{(x, \, t) \in Q_{r/ 2}(x_0, \, t_0)}
|U(x, \,t) - L^\pm_0(x)|^2 \, d \LLL(x, \, t) \le
C \left({r\over s}\right)^{2(1+\alpha)}\aint_{Q_s(x_0, \,
t_0)}U^2 \, d\LLL.$$
\smallskip\noindent
(b) Corresponding  to each $(x_1, \, t_1) \in \partial \Xi^\pm$ 
there is a unique polynomial function $L^\pm_1 \colon \R^n \to \R$ of
degree one with
$$|L^\pm_1(x_1)| + \|\nabla L^\pm_1(x_1)\| \le C$$
such that for each $0 < r \le 4$
$$\aint_{(x, \, t) \in Q_r(x_1, \, t_1)
\cap \Xi^\pm} |U(x, \, t) - L^\pm_1(x)|^2 \, d \LLL(x, \, t)
\le C \, r^{2(1 + \alpha)}.$$
\smallskip\noindent
(c) If  $(x_0, \, t_0) \in \Xi^\pm$ 
and $(x_1, \, t_1) \in \partial \Xi^\pm$ with
$$s = \delta_\star\big((x_0, \, t_0), \, (x_1, \, t_1)\big)
= \delta_\star\big((x_0, \, t_0), \, \partial \Xi^\pm\big)<
\infty$$
and if $L^\pm_1 \colon \R^n \to \R$ is the unique
polynomial function of degree one given in~(b)
associated with $(x_1, \, t_1)$
then the following in true: There is a polynomial function
$L^\pm_0 \colon \R^n \to \R$ of degree one with
$$|L^\pm_0(x_0)| + \|\nabla L^\pm_0(x_0)\|\, s
\le C \left(\aint_{Q_{s/ 2}(x_0, \, t_0)} \big(U - L^\pm_1\big)^2 \, d
\LLL\right)^{1 \over 2} + C\, s^{1 + \alpha}$$
such that for each $0 < r \le s$
$$\eqalign{\aint_{(x, \, t) \in Q_{r/ 2}(x_0, \, t_0)}
|U(x, \,t) -& L^\pm_1(x)- L^\pm_0(x)|^2 \, d \LLL(x, \, t) 
\cr &
\le
C \left({r\over s}\right)^{2(1+\alpha)}
\left(\aint_{Q_s(x_0, \, t_0)}\big(U- L_1\big)^2 \, d\LLL
+ C\, s^{2(1 + \alpha)}\right).\cr}$$
\smallskip\noindent
(d) If $L^\pm \colon \R^n \to \R$ are any 
polynomial functions of degree one then
$$r^2\, \aint_{Q_r(x, \, t) \cap \Xi^\pm}
|\nabla U - \nabla L^\pm|^2 \, d \LLL
\phantom{XXXXXXXXXXX}
$$
$$\phantom{XXX}
\le C\left( \aint_{Q_{2r}(x, \, t) \cap \Xi} |U - L^\pm|^2 \, d \LLL
+ \|\nabla L^\pm(0)\|^2\, r^{2(1 + \alpha)}\right)$$
whenever $(x, \, t) \in \Xi^\pm \cup \partial \Xi^\pm$ and 
$0 < r \le 1$.  Note that the domain of integration of the
right hand integral is $Q_{2r}(x, \, t) \cap \Xi$ not $Q_{2r}(x,\,t)
\cap \Xi^\pm$.
Also, in the right hand integral 
$$L^\pm(x) = \cases{L^+(x)& if $x \in \Xi^+$\cr
\phantom{x} &\cr
L^-(x)& if $x \in \Xi^-$\cr}.$$
We now show that our assumptions (a), (b),
(c), (d) above imply the condition $(\ast)$.

In case $(x_1, \, t_1) \in \partial \Xi^\pm$ and $0 < r \le
1$ we infer from
(d) and (b) that
$$\eqalign{\aint_{Q_r(x_1, \, t_1)\cap \Xi^\pm} 
&\left|\nabla U - \overline{\nabla U}\right|^2 \, d \LLL
\cr &
\le
\aint_{Q_r(x_1, \, t_1)\cap \Xi^\pm} |\nabla U - \nabla L^\pm_1 |^2
\, d \LLL
\cr &
\le {C \over r^2}\left( \aint_{Q_{2r}(x_1, \, t_1)\cap
\Xi} |U - L^\pm_1|^2\, d \LLL
+ \|\nabla L^\pm_1(0)\|^2 r^{2(1 + \alpha)}\right)
\cr &
\le C \, r^{2 \alpha}.\cr}$$

In case $(x_0, \, t_0 ) \in \Xi^\pm$,
we pick $(x_1, \, t_1)\in \partial \Xi^\pm$ such that
$$s = \delta_\star\big( (x_0, \, t_0 ),\, (x_1, \, t_1) \big)
= \delta_\star\big( (x_0, \, t_0 ),\, \partial \Xi^\pm\big).$$
Then (b) implies the existence of $L^\pm_1$ such that
$$\aint_{Q_r(x_1, \, t_1)\cap \Xi^\pm} |U - L^\pm_1|^2\, d \LLL
\le C\, r^{2(1 + \alpha)}$$
for each $0 < r \le 4$.

Assuming $0 < r \le 1$ we use (d), (c), (b) and (a) to infer
the existence of $L^\pm_0$ with $\| L_0 \| \le C\, s^\alpha$ such that

\noindent
either $0 < 4r < s\ge 1$ and
$$\eqalign{& \aint_{Q_r(x_0, \, t_0)}
\left|\nabla U - \overline{\nabla U}\right|^2 \, d \LLL
\cr &
\phantom{XX}
\le
\aint_{Q_r(x_0, \, t_0)}
|\nabla U - \nabla L^\pm_1 - \nabla L^\pm_0 |^2 \, d \LLL
\cr &
\phantom{XX}
\le {C\over r^2}
\left(\aint_{Q_{2r}(x_0, \, t_0)} |U - L^\pm_1- L^\pm_0|^2\, d \LLL
+ \|\nabla L^\pm_1(0)+ \nabla L^\pm_0(0)\|^2\, r^{2(1+\alpha)}\right)
\cr &
\phantom{XX}
\le {C\over r^2}
\Bigg[
\left({4r \over s}\right)^{2(1 + \alpha)}
\left(
\aint_{Q_{s}(x_0, \, t_0)} |U - L^\pm_1|^2\, d \LLL
+C\, s^{2(1 + \alpha)}\right)
\cr & \phantom{XXXXXXXX}
+ \|\nabla L^\pm_1(0)+ \nabla L^\pm_0(0)\|^2\, r^{2(1+\alpha)}\Bigg]
\cr &
\phantom{XX}
\le C\, r^{2\alpha}
\cr}$$
or $0 < 4r < s< 1$ and
$$\eqalign{& \aint_{Q_r(x_0, \, t_0)}
\left|\nabla U - \overline{\nabla U}\right|^2 \, d \LLL
\cr &
\phantom{XX}
\le
\aint_{Q_r(x_0, \, t_0)}
|\nabla U - \nabla L^\pm_1 - \nabla L^\pm_0 |^2 \, d \LLL
\cr &
\phantom{XX}
\le {C\over r^2}
\left(\aint_{Q_{2r}(x_0, \, t_0)} |U - L^\pm_1- L^\pm_0|^2\, d \LLL
+ \|\nabla L^\pm_1(0)+ \nabla L^\pm_0(0)\|^2\, r^{2(1+\alpha)}\right)
\cr &
\phantom{XX}
\cr}$$
$$\eqalign{
&\le {C\over r^2}
\Bigg[
\left({4r \over s}\right)^{2(1 + \alpha)}
\left(
\aint_{Q_{s}(x_0, \, t_0)} |U - L^\pm_1|^2\, d \LLL
+C\, s^{2(1 + \alpha)}\right)
\cr & \phantom{XXXXXXXX}
+ \|\nabla L^\pm_1(0)+ \nabla L^\pm_0(0)\|^2\, r^{2(1+\alpha)}\Bigg]
\cr &
\phantom{XX}
\le {C\over r^2}
\Bigg[
\left({4r \over s}\right)^{2(1 + \alpha)}
\left(
\aint_{Q_{2s}(x_1, \, t_1)\cap \Xi^\pm} |U - L^\pm_1|^2\, d \LLL
+C\, s^{2(1 + \alpha)}\right)
\cr & \phantom{XXXXXXXX}
+ \|\nabla L^\pm_1(0) + \nabla L^\pm_0(0)\|^2\, r^{2(1+\alpha)}\Bigg]
\cr &
\phantom{XX}
\le {C\over r^2}
\left({2r \over s}\right)^{2(1 + \alpha)}
\left(C\, (2s)^{2(1 + \alpha)} + C\, s^{2(1 + \alpha)}\right)
+ \|\nabla L^\pm_1(0)+ \nabla L^\pm_0(0)\|^2\, r^{2(1+\alpha)}
\cr &
\phantom{XX}
\le C\, r^{2\alpha}
\cr}$$
or $4r \ge s$ and
$$
%\eqalign{
\aint_{Q_r(x_0, \, t_0)\cap \Xi^\pm}
%&
\left|\nabla U - \overline{\nabla
U}\right|^2 \, d \LLL
%\cr &
\le
\aint_{Q_r(x_0, \, t_0)\cap \Xi^\pm} |\nabla U - \nabla L^\pm_1 -
\nabla L^\pm_0|^2 \, d \LLL
$$
$$
%\cr &
\le {C \over r^2}
\Bigg(\aint_{Q_{2r}(x_0, \, t_0)\cap \Xi}
|U - L^\pm - L^\pm_1|^2\, d \LLL
%\cr & 
%\phantom{XXXXXXXXX}
+ \|\nabla (L^\pm_1) + \nabla (L^\pm_0)\|\, r^{2(1 + \alpha)}
\Bigg)
$$ 
$$
%\cr &
\le {C \over r^2}\Bigg(2\,\aint_{Q_{2r}(x_0, \, t_0)\cap\Xi}
|U - L^\pm_1|^2\, d \LLL
+ 2\,\aint_{Q_{2r}(x_0, \, t_0)\cap \Xi} | L^\pm_0|^2\, d \LLL
%\cr &
$$
$$
\phantom{XXXXXXXXX}
+ \|\nabla L^\pm_1(0) + \nabla L^\pm_0(0)\|^2\, r^{2(1 + \alpha)}
\Bigg)
$$
$$
%\cr &
\le {C \over r^2}\Bigg(2\aint_{Q_{4r}(x_1, \, t_1)\cap
\Xi} |U - L^\pm_1|^2\, d \LLL
%\cr &
%\phantom{XXXXXXXXX}
+ C\aint_{Q_{s/2}(x_0, \, t_0)\cap \Xi}
|U - L^\pm_1|^2\, d \LLL
%\cr & 
$$
$$
\phantom{XXXXXXXXX}
+ \|\nabla L^\pm_1(0) + \nabla L^\pm_0(0)\|^2\, r^{2(1 + \alpha)}
\Bigg)
$$
$$
%\cr &
\le C\, r^{2 \alpha} +
+ {C \over r^2} \aint_{Q_{s}(x_1, \, t_1)\cap \Xi}
|U - L^\pm_1|^2\, d \LLL
%\cr &
\le C\, r^{2 \alpha}.
%\cr}
$$

\bigskip
{\bf Step 2 (Local energy estimates).}
Suppose
\smallskip \noindent
(b) $\Sigma$ is a H\"older continuously differentiable
hypersurface in $\B^n(0, \, 1)$.
\smallskip \noindent
(c) $\gamma \colon\, \B^n(0, \, 1)\sim \Sigma \to [1, \, \infty)$ is
continuous and bounded; we also write $\gamma(x, \, t) =
\gamma(x)$.
\smallskip\noindent
(d) $\kappa \colon\, \B^n(0, \, 1)\sim \Sigma \to \hbox{\rm Hom}
\left(\R^n , \, \R^n\right)$ is H\"older continuous  with exponent
$\alpha$ in each component of its domain
such that, for each~$x$, $\kappa(x) =
\left\{\kappa_{ij}(x)\right\}$ is a symmetric $n\times n$ matrix
with $v \bullet \big<v, \, \kappa(x)\big> \ge |v|^2$ for each~$v$; we
also write $\kappa(x, \, t) = \kappa(x)$.
\smallskip\noindent
(e) $f \colon\, \B^n(0,\, 1)\sim \Sigma \to \R^n$ is H\"older continuous
in each component of its domain;
we also write $f(x, \, t) = f(x)$.
\smallskip\noindent
(f) $g \colon Q_1\sim \Sigma \times (-1, \, 0]
\to \R$ is continuous and bounded; in our
applications $g$ will be the zero function.
\smallskip\noindent
(g) $w \colon Q_1 \to \R$ is a weak solution
of the partial differential equation
$$\gamma(x){\partial w \over \partial t}(x, \, t)
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa_{ij}(x)\, {\partial w\over \partial
x_i}(x, \, t)\, + f^j(x) \right) + g(x, \, t). $$
By this we mean that $w$ belongs to $\C^0\big((-1, \, 0], \,
\H^1(\B^n(0, \, 1))\big)$ and that, for each test
function $\varphi$ in $\DD\big(Q_1\big)$ and each $-1 < a < b \le 0$,
$$0 = \int_{(x, \, t) \in \T \times (a, \, b)}
\Bigg(-\gamma(x){\partial \varphi \over \partial t}(x,\, t)\,w(x, \, t)
+\sum_{j=1}^n{\partial \varphi\over \partial x_j}(x, \, t)
\sum_{i=1}^n \left(\kappa_{ij}(x){\partial w\over\partial x_i}(x, \, t)
+ f^j(x)\right)$$
$$\phantom{XX}- \varphi(x, \, t)\, g(x, \, t)\Bigg)
\, d {\cal L}^{n+1}(x, \, t)
+ \int_{x \in \T} \gamma(x)\big(\varphi (x, \, b)\, w(x, \, b)
-\varphi(x,\,a)\, w(x, \, a)\big)\, d \LL x.$$
Then we have the inequality
$$\int_{Q_{1\over 2}}
\left({\partial w \over \partial t}\right)^2 + |\nabla w|^2 \, \LLL
\le C \int_{Q_1} w^2 + |f|^2 + g^2 \, d \LLL.$$
\medskip
{\bf Proof.} We denote by $\eta \colon Q_1 \to [0, \, 1]$
a smooth cutoff function  which equals $1$ in
$Q_{1\over 2}$ and vanishes outside $Q_{3\over 4}$.
We first obtain an estimate on $\nabla w$.
We compute formally (to be interpreted in the sense of
distributions) using integration by parts with respect to both
space and time variables, Schwarz's inequality,
and the general inequality 
$2 ab \le \epsilon^{2} a^2 + \epsilon^{-2} b^2$. 
$$\eqalign{0 
&
=
\int_{Q_1}
\eta^2 w \Bigg( \gamma\, {\partial w\over \partial t} - \sum_j{\partial
\over \partial x_j}\left( \sum_i\kappa_{ij} {\partial w \over
\partial x_i} + f^j\right) - g\Bigg)\, d \LLL
\cr &
= \int_{Q_1} {\partial \over \partial t} \left({\eta^2 \,
\gamma\, w^2 \over 2}\right)\, d \LLL
-\int_{Q_1}{\partial \eta \over \partial t}\, \eta \, \gamma
\, w^2\, d \LLL
\cr &
\phantom{XXXX}
+\int_{Q_1}\sum_{ij}{\partial (\eta \, w)\over \partial x_j}
\kappa_{ij}
{\partial (\eta \, w)\over \partial x_i} \, d \LLL
-\int_{Q_1}\sum_{ij}{\partial \eta \over \partial x_j}\, w
\kappa_{ij}
{\partial \eta\over \partial x_i}\, w \, d \LLL
\cr &
\phantom{XXXXXX}
+\int_{Q_1} \sum_j {\partial \eta \over \partial x_j}\, f^j
\, \eta\, w + \eta\, f^j\, {\partial(w \, \eta)\over \partial
x_j} - \eta^2 \, w \, g \, \, d \LLL
\cr &
\ge\int_{x \in {\bf B}^n(0, \, 3/4)}
{\eta^2(x, \, 0) \, \gamma(x)\, w^2(x, \, 0) \over 2}\, d \LL x
- C\int_{Q_1} w^2 \, d \LLL
\cr &
\phantom{XX}
+\int_{Q_1}| \nabla(\eta\, w)|^2 \, \LLL
- C\int_{Q_1} |f|^2  \, d \LLL
\cr &
\phantom{XXX}
- {1 \over 2}\int_{Q_1}| \nabla(\eta\, w)|^2 \, \LLL
- C \int_{Q_1} g^2 \, d \LLL
\cr} $$
We infer
$$\int_{Q_{1\over 2}} |\nabla w|^2 \, d \LLL
\le\int_{Q_1} |\nabla(\eta \, w)|^2 \, d \LLL
\le C \int_{Q_1}w^2 + |f|^2 + g^2\, d \LLL.$$
In order to control ${\partial w \over \partial t}$ we estimate
$$\eqalign{0 
&
=
\int_{Q_1} \eta^2 {\partial w \over \partial t}
\left( \gamma\, {\partial w \over \partial t} - \sum_j{\partial
\over \partial x_j}\left( \sum_i\kappa_{ij} {\partial w \over
\partial x_i} + f^j\right) - g\right) \LLL
\cr &
\ge \int_{Q_1} \eta^2\, \gamma \,
\left({\partial w \over \partial t}\right)^2 \, d \LLL
\cr &
\phantom{X}
+ \int_{Q_1}\Bigg[{1\over 2}{\partial \over \partial t}
\left(\eta^2 \sum_{ij}{\partial w \over \partial x_j}
\kappa_{ij} {\partial w \over \partial x_i}\right)
- {\partial \eta \over \partial t}\, \eta \sum_{ij}
{\partial w \over \partial x_j} \kappa_{ij}
{\partial w \over \partial x_i}\Bigg]\, d \LLL
\cr &
\phantom{XX}
- {1 \over 4} \int_{Q_1} \eta^2\, \left({\partial w \over
\partial t}\right)^2\, d \LLL
- C \int_{Q_1} \eta^2\, |\nabla w|^2\, d \LLL
- C \int_{Q_1}\eta^2\, g^2 \, d \LLL
\cr &
\phantom{XXX}
- {1 \over 4}\int_{Q_1} \eta^2\, \left({\partial w \over \partial
t}\right)^2 \, d \LLL
+ \int_{Q_1}\sum_j {\partial \over \partial x_j}\left( \eta^2 \,
{\partial w \over \partial t}\right)\,f^j\, d \LLL
\cr}$$
$$\eqalign{
&
\ge
{1 \over 2}\int_{Q_1} \eta^2 \,\gamma(x)\, 
\left({\partial w \over \partial t}\right)^2 \,d \LLL
\cr &
\phantom{X}
+ {1 \over 2}\int_{x \in {\bf B}^n(0, \, 3/4)}
\eta(x, \, 0)^2 \sum_{ij}{\partial w \over \partial x_j}(x, \,
0)\,
\kappa_{ij}(x)\, {\partial w \over \partial x_i}(x, \, 0)\, d \LL x
\cr &
\phantom{XX}
-C \int_{Q_1}\eta^2\, |\nabla w|^2 \, d \LLL
- C \int_{Q_1} g^2 \, d \LLL
+ \int_{Q_1}\sum_j {\partial \over \partial x_j}\left( \eta^2 \,
{\partial w \over \partial t}\right)\,f^j\, d \LLL
\cr &
\ge
{1 \over 2}\int_{Q_1} \eta^2 \,\gamma(x)\, 
{\partial w \over \partial t}^2 \,d \LLL
\cr &
\phantom{X}
+ {1 \over 2}\int_{x \in {\bf B}^n(0, \, 3/4)}
\eta(x, \, 0)^2 |\nabla w(x, \, 0)|^2 \, d \LL x
\cr &
\phantom{XX}
-C \int_{Q_1}\eta^2\, |\nabla w|^2 \, d \LLL
- C \int_{Q_1} g^2 \, d \LLL
+ \int_{Q_1}\sum_j {\partial \over \partial x_j}\left( \eta^2 \,
{\partial w \over \partial t}\right)\,f^j\, d \LLL
\cr}.$$
We estimate the final term above
$$\eqalign{ \int_{Q_1}& {\partial \over \partial x_j}\left( \eta^2 \,
{\partial w \over \partial t}\right)\,f^j\, d \LLL
\cr &
= \int_{Q_1} 2 \eta \, {\partial \eta \over \partial x_j} \,
{\partial w \over \partial t} \,
f^j \, d \LLL
+\int_{Q_1} \eta^2 \, {\partial \over \partial t}
\left({\partial w \over \partial x_j}\, f^j\right) \, d \LLL
\cr &
\ge -{1 \over 4} \int_{Q_1} \eta^2\, \left({\partial w \over
\partial t}\right)^2 \, d \LLL
-C \int_{Q_1}|\nabla \eta|^2 \, |f|^2 \, d \LL
\cr &
\phantom{XX}
+\int_{Q_1} \eta^2 \, {\partial \over \partial t}
\left({\partial w \over \partial x_j}\,f^j\right) \, d \LLL
\cr &
= -{1 \over 4} \int_{Q_1} \eta^2\, \left({\partial w \over \partial
t}\right)^2 \, d \LLL
-C \int_{Q_1}|f|^2 \, d \LLL
\cr &
\phantom{XX}
+\int_{x \in {\bf B}^n(0, \, 3/4)}\eta^2(x, \, 0) \,
{\partial w \over \partial x_j}(x, \, 0)\,f^j(x) \, d \LL x
- \int_{Q_1} 2 \eta \, {\partial \eta \over \partial t}
\, {\partial w \over \partial x_j}\, f^j\,  d \LLL
\cr &
\ge
-{1 \over 4} \int_{Q_1} \eta^2\, {\partial w \over \partial
t}^2 \, d \LLL
-C \int_{Q_1}|f|^2 \, d \LLL
\cr &
\phantom{XX}
-{1 \over 4 n^2}
\int_{x \in {\bf B}^n(0, \, 3/4)}\eta^2(x, \, 0) \,
|\nabla w (x, \, 0)|^2
\, d \LL x
\cr &
\phantom{XXX}
-C \int_{x \in {\bf B}^n(0, \, 3/4)}\eta^2(x, \, 0) \,
|f(x)|^2 \, d \LL x
- {1 \over 4 n^2} \int_{Q_1}  \eta^2 \, |\nabla w|^2 \,d \LLL
\cr}$$
We note that
$$\eta^2 |\nabla w|^2 = |\eta\, \nabla w + \nabla \eta\, w -
\nabla \eta\, w|^2\le 2|\nabla(\eta \,w)|^2 + 2 |\nabla
\eta|^2 w^2$$
and that
$$ \int_{x \in {\bf B}^n(0, \, 3/4)}|f(x)|^2 \, d \LL x =
\int_{(x, \, t) \in {\bf B}^n(0, \, 3/4) \times (-1, \, 0]}
|f(x, \, t)|^2 \, d \LLL(x, \, t) \le
\int_{Q_1} |f|^2\, d \LLL$$
and combine our estimates to obtain
$$\int_{Q_{1\over 2}} \left({\partial w \over \partial
t}\right)^2 \, d \LLL
\le\int_{Q_1}\eta^2\, \left({\partial w \over \partial t}\right)^2 
\, d \LLL\phantom{XXXXXXXXXXXXXXXXXXX}$$
$$\phantom{XXXXXXXX}
\le C \int_{Q_1} |\nabla w|^2 + C \int_{Q_1} |f|^2 + g^2 \, d \LLL
\le C \int_{Q_1} w^2 + |f|^2 + g^2 \, d \LLL.$$
\bigskip
{\bf Step 3 (Regularity in a special case).}
Suppose that the assumptions of Step~2 above hold except that
our hypothesis (b) there is specialized so that
\smallskip\noindent
(b$^\ast$) $\sigma(x) = 0$ for each $x$ so that our hypersurface
$\Gamma$ is part of the vector subspace $x_n = 0$.
Then
\smallskip\noindent
(1) The restriction of $u$ to each compact subset of $\T
\times (-2, \, 0]$ is Lipschitz.
\smallskip\noindent
(2) The restriction $u^+$  of $u$ to $\T^+ \cup \Gamma$ is infinitely
differentiable as
is the the restriction $u^-$ of $u$ to $\T^- \cup \Gamma$.
\smallskip\noindent
(3) There is a constant $C$ (independent of $u$) such that
$$|u(0,\, 0)|^2 + \|\nabla u^+(0, \, 0)\|^2 + \|\nabla u^-(0, \, 0)\|^2
\le C \int_{Q_1} u^2\, d \LLL.$$
\smallskip\noindent
(4) For each $0 < r < 1$,
$$ {1 \over r^4} \aint_{Q_r \cap \T^+}\big| u - u(0,\, 0) - 
\nabla u^+(0, \, 0) \big|^2 d \LLL \le C \int_{Q_1} u^2\, d \LLL$$
and
$${1 \over r^4} \aint_{Q_r \cap \T^-}\big| u - u(0,\, 0) - 
\nabla u^-(0, \, 0) \big|^2 d \LLL
\le C \int_{Q_1} u^2\, d \LLL.$$
\medskip
{\bf Proof.}
We infer from our local energy estimate in Step 2 that the
restriction of $u$ to the interior $\T_0$ of any compact subset of
$\T \times (-2, \, 0]$ has square summable first derivatives,
i.e. belongs to $\H^1(\T_0)$.  This implies that the
restriction of a strictly defined $u$ to $\Gamma \cap \T_0$ belongs to
belongs to $\H^{1 \over 2}(\Gamma \cap \T_0)$.
Since our interface is a subset of the hyperplane $x_n=0$
and our partial differential equation is linear we infer that
any partial derivative of $u$ which does not require
differentiation in the $x_n$ variable itself must be a
weak solution to our partial differential equation, hence
itself must belong to $\H^1(\T_0)$ so that restrictions to
$\Gamma$ belong to $\H^{1 \over 2}(\Gamma \cap \T_0)$.
We infer that the restriction of $u$ to $\Gamma$ must be
infinitely differentiable.  The assertions
above now follow from the parabolic Schauder estimates
as set forth in Theorem~10.1 on page~351 of~[LSU].
\bigskip
{\bf Step 4 (An approximation estimate).}
For each $\epsilon >0$ there is $\delta > 0$ with the
following property: Suppose $\gamma^\star$ and
$\kappa^\star$ denote what the functions $\gamma^0$ and $\kappa^0$
respectively of our original hypotheses would be
in case $\sigma =0$
and $\Gamma$ were part of the vector subspace $x_n = 0$.
Suppose also
\smallskip \noindent
(b$^\ast$) $\Sigma = Q_1 \cap \big\{ x \colon x_n =
0\big\}$
\smallskip\noindent
(c$^\ast$) $\gamma \colon Q_1 \sim \Sigma \to [1, \, \infty)$ is
continuous and bounded with
$$\|\gamma - \gamma^\star\|_{{\bf L}_2} \le \delta.$$
(d$^\ast$) $\kappa \colon Q_1 \sim \Sigma
\to \hbox{\rm Hom}\left(\R^n
, \, \R^n\right)$ is H\"older continuous  with exponent
$\alpha$ such that, for each~$x$, $\kappa(x) =
\left\{\kappa_{ij}(x)\right\}$ is a symmetric $n\times n$ matrix
with $v \bullet \big<v, \, \kappa(x)\big> \ge |v|^2$ for each~$v$.
Also
$$\|\kappa - \kappa^\star\|_{{\bf L}_2} \le \delta.$$
(e$^\ast$) $f \colon Q_1 \sim \Sigma \to \R^n$ is H\"older continuous
with
$$\|f \|_{{\bf L}_2} \le \delta.$$
(f$^\ast$) $g \colon Q_1\sim \Sigma \times (-2, \, 0] \to \R$
is continuous and bounded with
$$\|g \|_{{\bf L}_2} \le \delta.$$
(g$^\ast$) $w \colon Q_1 \to \R$ is a weak solution
of the partial differential equation
$$\gamma(x){\partial w \over \partial t}(x, \, t)
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa_{ij}(x)\, {\partial w\over \partial
x_i}(x, \, t)\, + f^j(x) \right) + g(x, \, t). $$
with
$$\int_{Q_1} u^2 \, d \LLL \le 1.$$
By this we mean in particular
that $w$ belongs to $\C^0\big((-1, \, 0], \,
\H^1(\B^n(0, \, 1))\big)$ and that, for each test function 
$\varphi$ in $\DD\big( Q_1\big)$ and each $-1 < a < b \le 0$,
$$\eqalign{0 
&
= \int_{(x, \, t) \in {\bf B}^n(0, \, 1) \times (a, \, b)}
\Bigg(-\gamma(x)
{\partial \varphi \over \partial t}(x,\, t)\,w(x, \, t)
\cr &
\phantom{XX}
+\sum_{j=1}^n{\partial \varphi\over \partial x_j}(x, \, t)
\sum_{i=1}^n\left(\kappa_{ij}(x)
{\partial w\over\partial x_i}(x, \, t) + f^j(x)\right)
- \varphi(x, \, t)\, g(x, \, t)\Bigg)
\, d {\cal L}^{n+1}(x, \, t)
\cr &
\phantom{XXXX}
+ \int_{x \in \T} \gamma(x)\big(\varphi (x, \, b)\, w(x, \, b)
-\varphi(x,\,a)\, w(x, \, a)\big)\, d \LL x.\cr}$$
Then there is a function $v \colon Q_{1 \over 2} \to \R$
which  is a weak solution of  the partial  differential
equation
$$\gamma^\star(x){\partial v \over \partial t}(x, \, t)
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa^\star_{ij}(x)\, {\partial v\over \partial
x_i}(x, \, t) \right) $$
with
$$\int_{Q_{1/2}}|u - v|^2 \, d \LLL < \epsilon.$$
\medskip
{\bf Proof.}  The negation of our assertion implies the
existence of
$\{\gamma_k\}_k$, $\{\kappa^k\}_k$, $\{f_k\}_k$, $\{g_k\}_k$, 
$\{w_k\}_k$  for $k = 1,\, 2, \, 3, \, \ldots$ (with  the obvious
meaning) such  that
$$\|\gamma_k-\gamma^0\|_{{\bf L}^2(\T)} \to 0,\quad
\|\kappa^k-\kappa^0\|_{{\bf L}^2(\T)} \to 0,\quad
\|f_k\|_{{\bf L}^2(\T)} \to 0,\quad
\|g_k\|_{{\bf L}^2(\T)} \to 0$$
as $k \to \infty$ while for each $k$ 
$$\int_{Q_1} w_k^2 \, d \LLL \le 1$$
and $w_k$ weakly solves the partial differential equation
$$\gamma_k(x){\partial w_k \over \partial t}(x, \, t)
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa^k_{ij}(x)\, {\partial w_k\over \partial
x_i}(x, \, t)\, + f_k^j(x) \right) + g_k(x, \, t)$$
and 
$$\int_{Q_{1/2}}|w_k - v|^2 \, d \LLL \ge \epsilon$$
for every function $v$ which is a weak solution of the
partial differential equation
$$\gamma^0(x){\partial v \over \partial t}(x, \, t)
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa^0_{ij}(x)\, {\partial v\over \partial
x_i}(x, \, t) \right). $$
Our interior estimates in Step~2 guarantee that the $\H^1$
norms of the $w_k$'s are uniformly  bounded in $Q_{1 \over
2}$.  This implies the existence of a function 
$w_0 \colon Q_{1\over 2} \to \R$ with square summable  first
derivatives together with a a subsequence
$\ell(1), \, \ell(2), \, \ell(3), \, \ldots$ of $1, \, 2, \,
3, \, \ldots$ such that (note B.7), as $i \to \infty$,
$w_{\ell(i)} \h w_0$ in $\H^1$ (weak convergence) while
$w_{\ell(i)} \to w_0$ in $\L_2$ (strong convergence).
This convergence is adequate to guarantee that $w_0$
satisfies the required  equation.
\bigskip
{\bf Step 5 (How to  flatten $\Gamma$).}
We now consider a change of coordinates
$$\psi(x)=
\bigg(x_1, \, x_2, \, \ldots, x_{n-1}, \, x_n-
\sigma(x_1, \, x_2, \, \ldots, x_{n-1})\bigg)$$
which carries our  hypersurface $\Gamma$ to the plane 
$x_n = 0$.
%We set $B= \psi(\T)$, $B^+= \psi(\T^+)$, $B^-= \psi(\T^-)$.
We check that the function 
$$U(x, \, t) = u ( \psi^{-1}(x), \, t)$$
satisfies the partial differential equation
$$\gamma^\ast(x){\partial U \over \partial t}(x, \, t)
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa^\ast_{ij}(x)\, {\partial U\over \partial
x_i}(x, \, t)\right) $$
where
$$\gamma^\ast = \gamma^0 \circ \psi^{-1}$$
$$\kappa^\ast_{k\ell} =
\sum_{ij}\left(\kappa^0_{ij}\circ \psi^{-1}\right)\left(
{\partial \psi^k\over \partial x_i}\circ \psi^{-1}\right)
\left({\partial \psi^\ell \over \partial x_j}\circ \psi^{-1}\right)$$
We also check that if $L\colon \R^n \to \R$ is a polynomial of degree
one with $\|\nabla L(0)\| \le M$ (the size of $M$ will be controlled
later) then the function $U^0 = U - L$ satisfies the partial
differential equation
$$\eqalign{\gamma^\ast(x){\partial U^0 \over \partial t}(x, \, t)
&
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa^\ast_{ij}(x)\left( {\partial U^0\over \partial
x_i}(x, \, t) + {\partial L \over \partial x_i}(0)\right)\right) 
\cr &
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa^\ast_{ij}(x) {\partial U^0\over \partial
x_i}(x, \, t)+ f^j(x)\right)\cr} $$
where
$$f^j(x) = \sum_{i=1}^n \kappa_{ij}^\ast(x)\, {\partial L
\over \partial x_i}(0).$$
We consider a new pointwise H\"older norm which associates to a
function $h(x, t)$ at $0$ the number
$$[h]_\alpha(0) = \inf_{h_0}\sup_{r \le 1}\left(\aint_{(x, \, t)\in
Q_r}\left| h(x, \, t) - h_0(x)\right|^2 \, d \LLL(x, \,
t)\right)^{1 \over 2}$$
where the infimum is taken over all functions $h_0(x)=
h_0(x, \, t)$ which are constant when $x_n >0$ and also when
$x_n < 0$.
We infer from our original hypothesis~(b) and our calculation
above (since $M$ is assumed to be fixed) that at $0$
$$[\kappa^\ast_{ij}]_\alpha + [ f^j]_\alpha \le C \, \| \sigma \|_{C^{1,
\alpha}}.$$
If we suitably rescale our coordinate system then the new
$[\kappa^\ast_{ij}]_\alpha$ and $[f^j]_\alpha$ 
can be made arbitrarily small.
\bigskip
{\bf Step 6 (A decay estimate).}
Suppose $C_0$ is  the constant of Step~3 (called $C$ there) 
and that $0 < r < 1/2$ is fixed with
$$4\, C_0 \, r^{2(1-\alpha)} < 1.$$ 
Pick and fix $\epsilon_0>0$ such that
$$4 \, \epsilon_0^2 < r^{n + 4 + 2 \alpha}.$$
Now choose $\delta_0 > 0$ such that
if the hypotheses of Step~4 hold with $\delta$ there
replaced by $\delta_0$, $u$ there replaced by
$W$, and $v$ there replaced by $V$, then
$$\int_{Q_r}\big|V - W|^2\, d \LLL \le \epsilon_0^2.$$
We assert the existence of $C < \infty$ with the
following property.
Suppose $U$ [resp. $U^0$] is as in Step~5 with
$$\int_{Q_1}U^2\, d \LLL \le 1 \quad \left[\hbox{resp. }
\quad \int_{Q_1}(U^0)^2\, d \LLL \le 1\right]$$
and
$$[\kappa^\ast_{ij}]_\alpha \le {1 \over n^2} \delta_0 
\quad\left[ \hbox{resp. } \quad
[\kappa^\ast_{ij}]_\alpha\le {1 \over n^2} \delta_0 \hbox{
and } [f^j]_\alpha \le {1 \over n} \delta_0\right]$$
for each $i$ and $j$.
Then there exist constant functions
$$a_0(x)=0, \, a_1(x),
\, a_2(x), \, \ldots \quad\hbox{with} \quad
\big|a_k(x) - a_{k+1}(x)\big| \le C\, r^{k ( 1 +
\alpha)}$$
for each $k$ and $x$
together with piecewise constant vector valued functions
$$A_0(x)=0, \, A_1(x), \, A_2(x), \, \ldots \quad\hbox{with}
\quad
\big|A_k(x)- A_{k+1}(x)\big| \le C\, r^{k \alpha}$$
for each $k$ and $x$
whose values are constant vectors in $\R^n$  when $x_n > 0$ and
also constant vectors in $\R^n$ when $x_n < 0$
such that
$$\aint_{(x, \, t) \in Q_{r^k}}\big|U(x, \, t) - a_k(x )
- A_k(x) \bullet x \big|^2 \, d \LLL(x, \, t) \le r^{2k(1 + \alpha)}$$
$$\left[\hbox{resp. } \quad
\aint_{(x, \, t) \in Q_{r^k}}\big|U^0(x, \, t) - a_k(x )
- A_k(x) \bullet x \big|^2 \, d \LLL(x, \, t) \le r^{2k(1 + \alpha)}
\right]$$
for each  $k = 0, \, 1, \, 2, \, \ldots$.
\medskip
{\bf Proof.} We consider only the case of the function $U^0$.
For $k=0$ we take $a_0(x) = 0$, $A_0(x) = 0$ as indicated
so that our assertion holds by assumption.  Assume then that our
assertion holds up to
including some nonnegative integer~$k$ and set
$$W(x, \, t) = {1 \over r^{k(1 + \alpha)}} \Big(
U^0\big(r^k x, \, r^{2k}t\big) - a_k\big(r^k x\big)
- A_k\big(r^k x\big)\bullet x\Big).$$
Then
$$\int_{Q_1} |W|^2 \, d \LLL \le 1$$
and we use Step~5 to infer that
$W$ satisfies the partial differential equation
$$\eqalign{\gamma^\star(x)
&
{\partial W \over \partial t}(x, \, t)
\cr &
= \sum_{j=1}^n{\partial \over \partial x_j}
\left(\sum_{i=1}^n \kappa^\star_{ij}(x)\, {\partial W\over \partial
x_i}(x, \, t) + {1 \over r^{k \alpha}} \big(\kappa^\star_{ij}(x) -
\kappa^\diamond_{ij}(x)\big)\bullet A^i_k(x)
+{1 \over r^{k \alpha}} f_\star(x)
\right)\cr}$$
where
$$\gamma^\star(x) = \gamma^\ast\big(r^kx\big)\quad\hbox{and}\quad
\kappa^\star(x) = \kappa^\ast\big(r^kx\big)\quad\hbox{and}\quad
f^j_\star(x) = f^j(\big(r^kx\big) - \sum_{i=1}^n
\kappa^\diamond_{ij}\big(r^k x\big) \, {\partial L \over
\partial x_i}(0)$$
and $\kappa^\diamond$ equals $\kappa^0$ specialized to the
case in which $\sigma =0$ (and the hyperplane $\Gamma$ lies
in the vector subspace $x_n=0$).  We infer from Step~4 and
our choices above the
existence of a function $V(x, \, t)$ with
$$\int_{Q_r}|V - W|^2 \, d \LLL < \epsilon_0^2$$
which is a solution to the partial differential equation
$$\gamma^0(x){\partial V \over \partial t}(x, \, t)
= \sum_{j=1}^n{\partial \over \partial x_i}
\left(\sum_{i=1}^n \kappa^\star_{ij}(x)\, {\partial V\over \partial
x_i}(x, \, t) \right)$$
associated with the  specialization of our original
hypothesis (b) to the case $\sigma =0$.
According to Step~3 our function $V$ is smooth for $x_n \ge
0$ and also smooth for $x_n \le 0$.
We set $a_\ast(x) = V(0, \, 0)$ and define
$A_\ast(x)$ by requiring
$$A_\ast(x) \bullet x =
\cases{\big<x, \,  \nabla ^+V(0, \, 0)\big> & if $x_n > 0$\cr
\phantom{x} & \cr
\big<x, \,  \nabla ^-V(0, \, 0)\big> & if $x_n < 0$\cr}$$
and use Step~3 to infer
$$|V(0,\, 0)|^2 + \|\nabla V^+(0, \, 0)\|^2 + \|\nabla V^-(0, \, 0)\|^2
\le C_0 \int_{Q_1} V^2\, d \LLL.$$
and
$$\eqalign{\int_{(x, \, t) \in Q_r}\big|W(x, \, t)& - a_\ast(x) -
A_\ast(x) \bullet x \big|^2 \, d \LLL
\cr &
\le 2 \epsilon^2 +
2 \int_{(x, \, t) \in Q_r}\big|V(x, \, t) - a_\ast(x) -
A-\ast(x) \bullet x \big|^2 \, d \LLL
\cr &
\le 2 \epsilon^2 +
2\,C_0 \, r^4 \cdot r^{n+2}
\cr &
\le r^{2(1 + \alpha)}\cdot r^{n+2}
\cr}$$
We conclude that
$$\aint_{(x, \, t) \in Q_{r^{k+1}}}\bigg|U^0(x, \, t) - 
\big(a_k(x ) + r^{k(1 + \alpha)}a_\ast(x)\big)
-\big( A_k(x) + r^{k \alpha} A_\ast(x)\big) \bullet x \bigg|^2 \, 
d \LLL(x, \, t)$$
$$
\phantom{XXXXXXXXX}
\le r^{2k(1 + \alpha)}\cdot r^{2(1 + \alpha)}
= r^{2(k+1)(1 + \alpha)}$$
as required.
\bigskip
{\bf Step 7 (Corollary).}
Suppose $U(x, \, t)$ is as in Step~5.  Then there
is a constant function
$a_\star(x)$
and a piecewise constant vector valued function
$A_\star(x)$
whose value is a constant vector in $\R^n$  when $x_n > 0$ and
also a constant vector in $\R^n$ when $x_n < 0$ such
that
$$\sup_{0 < r < 1} {1 \over r^{2(1+ \alpha)}}
\aint_{(x, \, t) \in Q_r}
\big| U(x, \, t) - a_\star(x) - A_\star(x) \bullet x \big|^2
\, d \LLL(x,\, t)\le C \int_{Q_1} U^2 \, d \LLL$$
with
$$\big|a_\star(x)\big|^2 + \big|A_\star(x)\big|^2 \le
C \, \int_{Q_1}U^2 \, d \LLL;$$
here the value of the constant $C$ depends only on the value
of $[\kappa^\ast_{ij}]_\alpha$.
\medskip
{\bf Proof.} Use parabolic scaling to satisfy the condition
$$[\kappa^\ast_{ij}]_\alpha \le {1 \over n^2}
\delta_0 \quad \hbox{for
each $i$ and $j$}$$
of Step~6, and absorb the errors in the constant $C$.
\bigskip
\bigskip
{\bf Step 8 (Corollary).}
Suppose $U^0(x, \, t)$ is as in Step~5.  Then there
is a constant function
$a_\star(x)$
and a piecewise constant vector valued function
$A_\star(x)$
whose value is a constant vector in $\R^n$  when $x_n > 0$ and
also a constant vector in $\R^n$ when $x_n < 0$ such
that
$$\eqalign{\sup_{0 < r < 1} {1 \over r^{2(1+ \alpha)}}
\aint_{(x, \, t) \in Q_r}
\big| U^0(x, \, t)
&
- a_\star(x) - A_\star(x) \bullet x \big|^2
\, d \LLL(x,\, t)
\cr &
\le C\left( \int_{Q_1} (U^0)^2 \, d \LLL + \sum_{j=1}^n [f^j]_\alpha^2
\right)}$$
with
$$\big|a_\star(x)\big|^2 + \big|A_\star(x)\big|^2 
\le C\left( \int_{Q_1} (U^0)^2 \, d \LLL + \sum_{j=1}^n [f^j]_\alpha^2
\right)$$
here the value of the constant $C$ depends only on the value
of $[\kappa^\ast_{ij}]_\alpha$.
\medskip
{\bf Proof.} Dominate $M$ in Step~5 by $\|A_\star(x)\|$ in
Step~7.  Then  apply Step~6 to the function
$$U^1 = { \delta_0 \, U^0(x, \, t) \over \left( \int_{Q_1}
(U^0)^2 \, d \LLL + \sum_{j=1}^n [f^j]_\alpha(0)]^2 \right)^{1
\over 2}}$$
which implies that the ``$f^j$'' associated with $U^1$ has
H\"older norm  less than $\delta_0$.
Use parabolic scaling.
\bigskip
{\bf Step 9 (Final estimates).}
Our object is to show that the asserted regularity of
our original function $u$ across the interface $\Gamma$.
In view of the change of coordinates $\psi$ in Step~5 which
flattened $\Gamma$, it is sufficient to show that the
perturbed function $U$ 
satisfies the four conditions (a), (b), (c), (d) of 
Step~1 when $\Xi$ there equals either
$\psi(\T^+)\cap \B^n(0, \, 1) \times (-1, \, 0]$
or
$\psi(\T^-)\cap \B^n(0, \, 1) \times (-1, \, 0]$
so that Campanato's embedding theorem will imply the
necessary regularity of $U$.
Away from the interface $x_n = 0$ the function $U$ is
H\"older continuously differentiable (since $u$ is
infinitely differentiable away from $\Gamma$)
so that the relevant estimates
hold immediately.  The main question is about the behavior
of $U$ near $\Gamma$.

Condition~(b) in Step~1 follows from Step~7 applied to the
function $U$.  Condition~(a) follows from Step~7 applied to
the function $U$
since a situation with no interface
discontinuities is a special case of Step~7, and we can scale
the estimate there to replace $Q_1$ by $Q_s$. 
Condition~(c) follows from Step~8 applied to the function $U^0$
since a situation with no interface
discontinuities is a special case of Step~8, and we can scale
the estimate there to replace $Q_1$ by $Q_s$. 
Condition~(d) follows from Step~2.
\E
\bigskip
\centerline{\larger 6. Sufficient conditions for crystal
positions}

\centerline{\larger and temperature fields to be close}
\bigskip
In the second conclusion of Theorem~5.2 we showed that large
space Dirichlet integrals of temperature implied rapid
decrease in our total energy functions $\EE_j(t)$.
In Theorem~5.3 we showed the essential {\sl a priori} 
Monge-Kantorovich H\"older continuity in time of heat 
distributions in approximating evolutions.
In this chapter we set forth the principal estimates which
enable us to control changes in crystal positions and in
temperature fields in terms of temperature Dirichlet integrals
and heat distribution changes.
\bigskip
{\bf 6.1 Theorem (Crystal positions, heat distributions, 
and Dirichlet integrals).}
{\sl
Suppose the following conditions hold
\smallskip\noindent
(a) $\l K\r$ and $\l L\r$ are crystals in $\KK$ with
$\LL(K \tt L) > 0$.
\smallskip\noindent
(b) $Q$ is a single heat distribution in $\QQ$.
\smallskip\noindent
(c) $T_K = c_{{}_K}^{-1} Q$, $T_L = c_{{}_L}^{-1} Q$ are associated
temperature fields.
\smallskip\noindent
(d) $U_K = 1/T_K$ and $U_L = 1/T_L$ are associated reciprocal
temperature fields.
\smallskip\noindent
Then
$$
\dir(U_K) + \dir(U_L) + \dir(T_K) + \dir(T_L) = \infty.$$
}
PROOF. The proof is in three parts.
\smallskip 
{\bf Part 1 (Alternatives across detectable interfaces).}
Suppose the following conditions hold
\smallskip\noindent
(a) $K$ and $L$ are sets in $\RR$.
\smallskip\noindent
(b) $Q\colon \RR \to \R^+$ is a positive function (a single ``heat distribution'').
\smallskip\noindent
(c) $T_K = c_{{}_K}^{-1} Q$ and $T_L = c_{{}_L}^{-1} Q$ (are associated
temperature fields); here.
$c_{{}_K}$ and $c_{{}_L}$ are defined by analogy with 2.5.
\smallskip\noindent
(d) $U^K = T_K^{-1}$ and $U^L = T_L^{-1}$ (are associated reciprocal
temperature fields).
\smallskip\noindent
(e) $p$ and $q$ are points in $\RR$.
\smallskip\noindent
Then one of the following two alternatives holds.
\smallskip\noindent
{\bf First alternative.}

\noindent
EITHER
\smallskip\noindent
(1) Based on information at $p$ and $q$, neither $K$ nor $L$ has a detectable
interface between $p$ and $q$, viz.
$$\bigg(\{p, \, q\} \subset K \hbox{\rm \ or } \{ p, \, q\} \subset \RR \sim K\bigg)
\quad \hbox{and} \quad
\bigg(\{p, \, q\} \subset L \hbox{\rm \ or } \{ p, \, q\} \subset \RR \sim L\bigg).$$
OR
\smallskip\noindent
(2) Based on information at $p$ and $q$, both $K$ and $L$ detect interfaces
between $p$ and $q$ and they are of the same type, viz.
$$\bigg( p \in K \cap L \hbox{ \rm \ and }q \in (\RR \sim K) \cap (\RR \sim L)\bigg)
\phantom{XXXXXXXXXX}$$
$$\phantom{XXXXXXXXXX}\hbox{or}\quad
\bigg( q \in K \cap L \hbox{ \rm \ and }p \in (\RR \sim K) \cap (\RR \sim L)\bigg).$$
\medskip\noindent
{\bf Second alternative.}
$$\eqalign{
\big| U_K(p) - U_K(q) \big| + \big| U_L(p) - U_L(q) \big| +
 \big| T_K(p) - T_K(q) \big| 
&+ \big| T_L(p) - T_L(q) \big|
\cr &
\ge 2 \left( c_{{}_s} c_{{}_\ell}\right)^{1 \over 2}\left( {1 \over
c_{{}_s} } - {1 \over c_{{}_\ell}}\right).\cr}$$
\bigskip
{\bf Proof of Part 1.}  The proof of the assertions of Part~1 is in two steps

{\bf Step 1.}  We assert
\smallskip\noindent
(1) For any three numbers $A$, $B$, $C$ we have the inequality
$$|A- B| + |A- C| \ge |B - C|.$$
Indeed, if $C \le B$, then
$$|B-C| = B-C = B-A + A -C \le |B-A| + |A-C|.$$
\smallskip\noindent
(2) For any three positive numbers $A$, $B$, $R$ we have the inequality
$$AR + B/R \ge 2 \sqrt{AB}.$$
Indeed, the function $f(R) = AR + B/R$ has its unique positive minimum
when $f'(R) = A - B/R^2 = 0 $, $R = \sqrt{B/A}$, at which point the function
takes value 
$$A \sqrt{B/A} + B/\sqrt{B/A} = 2 \sqrt{AB}.$$
\bigskip
{\bf Step 2.}  We will assume the first alternative does not hold and will show
that the second alternative must then hold.  There is thus
a detectable interface between $p$ and $q$.  For the purpose of our proof
it is sufficient to treat the case in which $p \in K$, $q \in \RR \sim K$, which
assumption we make.  
Other cases follow by various interchanges of $K$ with $L$ and/or of $p$ with $q$.  
There are then three cases.

Case 1. $p, \, q \in L$.

Case 2. $p, \, q \in \RR \sim  L$.

Case 3. $p \in \RR \sim L, \, q \in L$.

\noindent
We abbreviate $a = 1/c_{{}_s} > b = 1/c_{{}_\ell}$ and
$R = Q(p)$, $S = Q(p)$ and consider our three cases.
\smallskip\noindent
Case 1.  We use Step 1 to estimate
$$\eqalign{\big|a R - b S\big| + \big|a R - a S\big| 
& + \big|1/a R -1/  b S\big| + \big|1/a R -1/  a S\big|
\cr & \ge (a-b)S + (1/b - 1/a)/S
\cr & = (a-b)S + (a-b)/abS
\cr & \ge 2 (a-b)/ \sqrt{ab}
\cr }$$
\smallskip\noindent
Case 2.  We use Step 1 to estimate
$$\eqalign{\big|a R - b S\big| + \big|b R - b S\big| 
& + \big|1/a R -1/  b S\big| + \big|1/b R -1/  b S\big|
\cr & \ge (a-b)R + (1/b - 1/a)R
\cr & = (a-b)R + (a-b)/abR
\cr & \ge 2 (a-b)/ \sqrt{ab}
\cr }$$
\smallskip\noindent
Case 3.  We use Step 1 to estimate
$$\eqalign{\big|a R - b S\big| &+ \big|b R - a S\big| 
 + \big|1/a R -1/  b S\big| + \big|1/b R -1/  a S\big|
\cr & 
= (a/b) \big| bR - (b^2/a)S \big| + \big|bR - aS\big|
+ \big|1/aR - 1/bS \big| + (a/b)\big|1/aR - b/a^2S\big|
\cr &
\ge  \big| bR - (b^2/a)S \big| + \big|bR - aS\big|
+ \big|1/aR - 1/bS \big| + \big|1/aR - b/a^2S\big|
\cr &
\ge (a - b^2/a)S + (1/b - b/a^2)/S
\cr &
= \big((a^2 - b^2)/a\big)S + \big((a^2 - b^2)/ab^2\big)/S 
\cr &
\ge 2(a^2 - b^2)/ab
\cr &
\ge 2(a-b)/\sqrt{ab}
\cr }$$
since $(a+b)/\sqrt{ab} > 1$. 
\bigskip
{\bf Part 2 (Interfaces and Dirichlet's integrals).}
Corresponding to each (large) number $M$
there is a (small) positive number $\delta$ with with the following property.
Suppose the conditions below are satisfied.
\smallskip\noindent
(a) $K$ and $L$ are $\LL$ measurable sets in $\RR$.
\smallskip\noindent
(b) $Q\colon \RR \to \R$ belongs to $\L_2(\RR)$ and $1/Q$ also belongs to
$\L_2(\RR)$ (a single ``heat distribution'').
\smallskip\noindent
(c) $T_K = c_{{}_K}^{-1}, Q$ $T_L = c_{{}_L}^{-1} Q$ (are associated
temperature fields); here.
$c_{{}_K}$ and $c_{{}_L}$ are defined by analogy with 2.5.
\smallskip\noindent
(d) $U^K = T_K^{-1}$ and $U^L = T_L^{-1}$ (are associated reciprocal
temperature fields).
\smallskip\noindent
(e) $A = \B^n(0, \, 1) \cap\{ x \colon x_n > 0 \}$.
\smallskip\noindent
(f) $B = \B^n(0, \, 1) \cap\{ x \colon x_n < 0 \}$.
\smallskip\noindent
(g) $A_K = A \cap K$, $A_L = A \cap L$, $B_K = B \cap K$, $B_L = B \cap L$,
\smallskip\noindent
(h) $\LL\big(A \sim A_K\big) + \LL\big(B_K\big) < \delta$.
\smallskip\noindent
(i) One of the following three possibilities holds
\smallskip
(i.1) $\LL\big( A \sim A_L\big)+\LL\big( B \sim B_L\big) < \delta;$
\smallskip
(i.2) $\LL\big( A_L\big)+\LL\big(B_L\big) < \delta;$
\smallskip
(i.3) $\LL\big( A_L\big)+\LL\big( B \sim B_L\big) < \delta.$
\smallskip\noindent
Then
\smallskip\noindent
$$\eqalign{
\dir\big(U_K; \B^n(0, \, 1)\big) 
+ \dir\big(U_L; \B^n(0, \, 1)\big) 
& +\dir\big(T_K; \B^n(0, \, 1)\big)
\cr & + \dir\big(T_L; \B^n(0, \, 1)\big) \ge M. \cr}\leqno{(1)}$$
\smallskip\noindent
Furthermore, if $0 < r < 1$ and
$$U_K^r(x) = U_K(x/r), \,
U_L^r(x) = U_K(x/r), \,
T_K^r(x) = T_K(x/r), \,
T_L^r(x) = T_L(x/r),$$
for each $x$, then 
\smallskip\noindent
$$\eqalign{
\dir\big(U^r_K; \B^n(0, \, 1)\big) 
& + \dir\big(U^r_L; \B^n(0, \, 1)\big) +
+\dir\big(T^r_K; \B^n(0, \, 1)\big) 
\cr &
+ \dir\big(T^r_L; \B^n(0, \, 1)\big) \ge Mr^{n-2}. \cr } \leqno{(2)}$$
\bigskip
{\bf Proof of Part 2.}  The proof of Part 2 is in two steps.

{\bf Step 1.}  Suppose $f(t)$ is a function with square summable first derivatives
for $|t| \le \epsilon$ and there exist $-\epsilon \le s < t \le \epsilon$ such
that
$$|f(s) - f(t)| \ge 
{1 \over 2} \left( c_{{}_s} c_{{}_\ell}\right)^{1 \over 2}\left( {1 \over
c_{{}_s} } - {1 \over c_{{}_\ell}}\right).$$
Then
$$\int_{[s, \, t]}(f')^2 \, d {\cal L}^1
\ge
{1 \over \epsilon} { c_{{}_s} c_{{}_\ell}\over 8}\left( {1 \over
c_{{}_s} } - {1 \over c_{{}_\ell}}\right)^{1 \over 2}.$$
Indeed
$$|f(s) - f(t)| \le \int_{[s, \, t]}|f'|^2 \, d {\cal L}^1
\le (t-s)^{1 \over 2}\left( \int_{[s, \, t]}(f')^2 \, d {\cal L}^1\right)^{1 \over 2}.$$
\medskip
{\bf Step 2.}  If any of the Dirichlet integrals in first 
conclusion are infinite,
the first conclusion is immediate.  We therefore assume that each of
these
Dirichlet integrals is finite and further require that $U_K$, $U_L$, 
$T_K$, $T_L$
be strictly defined at all Lebesgue points.  We infer that 
for ${\cal L}^{n-1}$
almost every $y$ in $\B^{n-1}(0, \, 1)$ the functions
$$ s \mapsto U_K(y, \, s), s \mapsto U_L(y, \, s), 
s \mapsto T_K(y, \, s), s \mapsto T_L(y, \, s)$$
are absolutely continuous with
square summable first $s$ derivatives for $|y|^2 + s^2 < 1$ 
[MC, Lemma 3.1.1].  According
to Fubini's theorem
$$DIR\big(U_K; \B^n(0, \, 1)\big) \ge \int_{y \in {\bf B}^{n-1}(0, \, 1)}
\int_{\{s \colon |y|^2 + s^2 < 1\}}
\left({\partial U_K \over x_n}(y, \, s)\right)^2 \, d{\cal L}^1 s \, d {\cal L}^{n-1} y,$$
with corresponding inequalities for $U_L$, $T_K$, $T_L$.  
Suppose then it is possibility (i.1) above which occurs and $\epsilon > 0$ is very small.
If $\delta$ is extremely small we infer from hypotheses (h) and (i.1) together
with Fubini's theorem that for most $y$'s in ${\bf B}^{n-1}(0, \, 1)$ there
will exist $p = (y, \, s) \in A_K \cap A_L$ and $q = (y, \, t) \in (\RR \sim K)
\cap L$ with $-\epsilon < s < t < \epsilon$.  We check that the first
alternative of  Part~1 does not hold for such $p$ and $q$
and infer from the second alternative that, for such $y$'s,
$$ \eqalign{
\big| U_K(p) - U_K(q) \big| + \big| U_L(p) - U_L(q) \big| +
 \big| T_K(p) - T_K(q) \big| 
\cr &+ \big| T_L(p) - T_L(q) \big|
\cr & \ge 2 \left( c_{{}_s} c_{{}_\ell}\right)^{1 \over 2}\left( {1 \over
c_{{}_s} } - {1 \over c_{{}_\ell}}\right).\cr }$$
Hence, at least one of the four numbers on the left hand side must be at least
as large as one fourth of the number on the right hand side.  Suppose, say,
for a particular $y$,
$$
\big| U_K(p) - U_K(q) \big|
\ge {1 \over 2} \left( c_{{}_s} c_{{}_\ell}\right)^{1 \over 2}\left( {1 \over
c_{{}_s} } - {1 \over c_{{}_\ell}}\right).$$
Then the estimate of Step~1 implies that
$$\int_{\{s \colon |y|^2 + s^2 < 1\}}
\left({\partial U_K \over x_n}(y, \, s)\right)^2 \, d{\cal L}^1 s
\ge
{1 \over \epsilon} { c_{{}_s} c_{{}_\ell}\over 8}\left( {1 \over
c_{{}_s} } - {1 \over c_{{}_\ell}}\right)^{1 \over 2}.$$
Corresponding estimates would hold for $U_L$, $T_K$, $T_L$.
The same argument would apply if it were possibility (i.2) or (i.3) which
held.
We infer the first conclusion of Part~2 from Fubini's theorem as indicated above.  The
second conclusion follows from the first.
\bigskip
{\bf Part 3.}  We now let $K$, $L$, $T_K$, $T_L$, $U_K$, $U_L$ be as in the
statement of the theorem.  The remainder of the proof the the theorem is in
two steps.

{\bf Step 1.}
We assert that $\partial \l K \r \not= \partial \l L \r$.  Indeed, if 
$\partial \l K \r =\partial  \l L \r$, then the Constancy Theorem
[FH 4.1.7] would
imply that $\l K \r - \l L \r = z \l \O \r$ for some integer $z$.  On the other hand, 
for $\LL$ almost every $x$ in $\O$,
$$z = \cases{
 1 & if $x \in K \sim L$\cr
           \phantom{x} & \cr
 -1 & if $ x \in L \sim K$\cr
           \phantom{x} & \cr
 0 & if $ x \in 
(K \cap L) \cup
(\O \sim K)\cap
(\O \sim L)$\cr}.$$
It follows that at most one of the three sets on the right hand side can have
positive measure.
Since $K \tt L = (K \sim L) \cup (L \sim K)$ has positive measure by assumption, 
the measure of
$(K \cap L) \cup
(\O \sim K)\cap
(\O \sim L)$
must vanish so that $K$ and $L$ together $\LL$ almost partition $\O$.
We infer from assumption~2.2.1 that either $\LL(K) > 0$ and
$\LL(L)> 0$ or $\LL(\O \sim K) > 0$ and $\LL(\O \sim L) > 0$.
In either case we further infer that $\partial \l K \r = -
\partial \l L \r \not= 0$ which implies our assertion.
\medskip
{\bf Step 2.} We utilize the structure of sets of finite perimeter set forth
in [FH 4.5.6] to infer (interchanging $K$ and $L$ if necessary)
that there
is a subset $J$ of $\partial K$ of positive $\HH$ measure such that, for
each $p$ in $J$, one of the following three alternatives holds:
either
$p \in L$ with $\Theta^n\left( \LL \rightangle L, \, p\right) = 1$,
or
$p \in \O \sim L$ with $\Theta^n\left( \LL \rightangle L, \, p\right) = 0$,
or
$p \in \partial L$ with ${\bf n}_L(p) = - {\bf n}_K(p)$.  For each fixed
such $p$ in $J$ and (large) number $M$ we then infer from Part~2
the existence of $r_0(p) > 0$ such that whenever $0 < r < r_0(p)$ then
$$\eqalign{\dir\big(U_K; \B^n(p, \, r)\big) + \dir\big(U_L; \B^n(p, \, r)\big) 
+ \dir&\big(T_K; \B^n(p, \, r)\big) 
\cr & + \dir\big(T_L; \B^n(p, \, r)\big) \ge M r^{n-2}.\cr}$$
Let $\B^n(p_1, \, r_1),\, \B^n(p_2, \, r_2),\, \B^n(p_3, \, r_3),\, 
\ldots$ be a covering of $J$ by closed balls
in the sense of the Besicovitch-Federer covering theorem [FH 2.8.14]
(with $\beta(n)$ equal to the number $2 \zeta +1$ there)
with $r_i < r_0(p_i)$ for each~$i$
We infer
$$\eqalign{
\dir(U_K) + \dir(U_L) + \dir(T_K) + \dir(T_L)
&
\ge {M \over \beta(n)} \sum_i r_i^{n-2}
\cr &
\ge {M \over \alpha(n) \beta(n) \sup_i r_i}\alpha(n)\sum_i r_i^{n-1}
\cr &
\ge {M \over \alpha(n) \beta(n) \sup_i r_i}\HH(J).
\cr }$$
The conclusion follows since both $M$ can be chosen arbitrarily large and
$\sup_i r_i$ can be chosen very small.  \E
\bigskip
{\bf 6.2 Corollary (Sufficient condition for two crystals to be close).}
{\sl
Corresponding to each (small) positive number $\epsilon$ and each
(large) number  $M$ there is a (small) positive number $\delta$ with
the following property.  Suppose
\smallskip\noindent
(a)  $\l K \r$ and $\l L \r$ are crystal in $\KK$.
\smallskip\noindent
(b) $P$ and $Q$ are heat distributions in $\QQ$ with $\big\| P - Q \big\| < \delta$.
\smallskip\noindent
(c) $S = c_{{}_K}^{-1} P$ and $T = c_{{}_L}^{-1} Q$ are associated
temperature fields.
\smallskip\noindent
(d) $U = 1/S$ and $V = 1/T$ are associated reciprocal
temperature fields.
\smallskip\noindent
(e) $\| U \|_{{}_{{\bf L}_2}} + 
\| V \|_{{}_{{\bf L}_2}} + 
\| S \|_{{}_{{\bf L}_2}} + 
\| T \|_{{}_{{\bf L}_2}} \le M$.
\smallskip\noindent
(f) $\dir(U) + \dir(V) + \dir(S) + \dir(T)  \le M$.
\smallskip\noindent
Then 
$$\LL(K \tt L) < \epsilon.$$
}
\bigskip
PROOF.
The negation of our assertion says that there exist some $\epsilon$ and $M$ such
that, no matter how small we pick $\delta$,
there will be $K$, $L$, $S$, $T$, $U$, $V$ satisfying the conditions of the
theorem above for which $\LL(K\tt L) \ge \epsilon$.  
For each $i = 1, \, 2, \, 3, \, \ldots$ let
$K_i$, $L_i$, $S_i$, $T_i$, $U_i$, $V_i$ be such a choice corresponding to 
$\delta_i = 2^{-i}$.  The compactness theorem for integral currents together
with Rellich's theorem guarantee the existence of a subsequence $j(1),\, j(2), \,
j(3), \, \ldots$ of $1,\, 2,\, 3,\, \ldots$ and limit (in the mass topology) 
crystals $\l K \r$, $\l L \r$ with $K \tt L \ge \epsilon$,
limit (in the strong $\L_2(\O)$ topology) temperature fields $S$, $T$ 
and inverse temperature fields $U$, $V$
with $\dir(U) + \dir(V) + \dir(S) + \dir(T)  \le M$, and hence
limit (in the strong $\L_2(\O)$ topology) heat distributions $P$, $Q$. 
Since $\lim_i \| P_i - Q_i \|_\ast = 0$ we infer $P = Q$.  This is incompatible
with Theorem~6.1 which proves the present corollary.  \E
\bigskip
{\bf 6.3 Proposition (Sufficient condition for two temperature fields
to be close).}
{\sl
Corresponding to each (small) positive number $\epsilon$ and each (large)
number $M$ there is a (small) number $\delta$ with the following property.
Suppose
\smallskip\noindent
(a) $\l K \r$ and $\l L \r$ are crystals in $\KK$ with
$\LL(K \tt L) < \delta$.
\smallskip\noindent
(b) $P$ and $Q$ are heat distributions in $\QQ$ with
$\big\| P - Q \big\|_\ast < \delta$.
\smallskip\noindent
(c)  
$S = c_{{}_K}^{-1} P$, and $T = c_{{}_L}^{-1} Q$, are associated temperature
fields.
\smallskip\noindent
(d)
$U = 1/S$, and
$V = 1/T$ are associated reciprocal temperature fields.
\smallskip\noindent
(e) $\| U \|_{{}_{{\bf L}_2}} + 
\| V \|_{{}_{{\bf L}_2}} + 
\| S \|_{{}_{{\bf L}_2}} + 
\| T \|_{{}_{{\bf L}_2}} \le M$.
\smallskip\noindent
(f) 
$\dir(U) + \dir(V) + \dir(S) + \dir(T) \le M$.
\smallskip\noindent
Then
\smallskip\noindent
$$\| U  - V \|_{{}_{{\bf L}_2}} + \| S - T \|_{{}_{{\bf L}_2i}} < \epsilon.$$
}
\bigskip
PROOF.
The negation of our assertion says that there exist some $\epsilon$ and $M$
such that, no matter how small we pick $\delta$,
there will be $K$, $L$, $P$, $Q$, $S$, $T$, $U$, $V$ satisfying the conditions of the
theorem above for which 
$$\| U  - V \|_{{}_{{\bf L}_2}} + \| S - T \|_{{}_{{\bf L}_2}} \ge \epsilon.$$
For each $i = 1, \, 2, \, 3, \, \ldots$ let
$K_i$, $L_i$, $P_i$, $Q_i$, $S_i$, $T_i$, $U_i$, $V_i$ be such a choice corresponding to 
$\delta_i = 2^{-i}$.  The compactness theorem for integral currents together
with the compactness theorem for heat distributions and
with Rellich's theorem guarantee the existence of a subsequence $j(1),\, j(2), \,
j(3), \, \ldots$ of $1,\, 2,\, 3,\, \ldots$ and limit (in the mass topology) 
crystal $\l K \r$, limit (in the $\| \cdot \|_\ast$ topology) heat distribution
$P$,
limit (in the strong $\L_2(\O)$ topology) temperature fields $S$, $T$ 
and inverse temperature fields $U$, $V$
with $\| U \|_{{}_{{\bf L}_2}} + \| V \|_{{}_{{\bf L}_2}} + \| S \|_{{}_{{\bf L}_2}} + 
\| T \|_{{}_{{\bf L}_2}} \le M$
and
$\dir(U) + \dir(V) + \dir(S) + \dir(T)  \le M$.
In particular, 
$$\lim_i \LL \big( K_{j(i)} \tt K \big) = 
\lim_i \LL\big( L_{j(i)} \tt  K\big) =  0\leqno{(i)}$$ 
and
$$\lim_i \big\|P_i - P \big\|_\ast = \lim_i \big\|Q_i - P \big\|_\ast = 0\leqno{(ii)}$$
and
$$\lim_i \| U_{j(i)}  - U \|_{{}_{{\bf L}_2}} 
= \lim_i \| V_{j(i)}  - V \|_{{}_{{\bf L}_2}} 
= \lim_i \| S_{j(i)}  - S \|_{{}_{{\bf L}_2}} 
= \lim_i \| T_{j(i)}  - T \|_{{}_{{\bf L}_2}} =0\leqno{(iii)}$$
and 
$$\| U  - V \|_{{}_{{\bf L}_2}} + \| S - T \|_{{}_{{\bf L}_2}} \ge \epsilon.\leqno{(iv)}$$
Conditions
$(i)$, $(ii)$, and $(iii)$ imply  $S = T 
= c_{{}_K}^{-1} P$ and $U= V = 1/T$
which is incompatible with $(iv)$.
\E
\bigskip
\centerline{\larger 7. Selection of subsequences of approximating evolutions}
\bigskip
The convergence of the crystals, heat distributions, and 
temperature fields
in our approximating evolutions to limits which satisfy the heat equation
and Gibbs-Thomson relations is fairly complicated.  We describe 
in this chapter how to obtain subsequences of approximating evolutions
for which crystals, heat distributions, and temperature fields
converge very strongly at almost every time while heat distributions
converge at every time to a limit which
varies H\"older continuously in the Monge-Kantorovich metric.
We verify that the heat equation with varying crystals is satisfied
for our limit.
\bigskip
{\bf 7.1 Subsequences and Cantor's diagonal process.}  
We denote by $J_0$ the identity function on the positive
integers, i.e. $J_0(i) = i$ for each $i =  1, \, 2, \, 3, \, \ldots$.
By a {\bf subsequence} of $J_0$ we mean any increasing function
from the positive integers to the positive integers.
In case $H$ and $J$ are subsequences of $J_0$, we say that $H$ is
{\bf subsequence} of $J$ if and only of the 
image of $H$ is a subset of the image of $J$.
If $J^1$, $J^2$, $J^3$, $\ldots$  are subsequences of
$J_0$ such that
$J^{i+1}$ is a subsequence of $J^i$ for each~$i$, then {\bf the
result of Cantor`s diagonal process} is the subsequence of $J_0$
having image $\big\{J^1(1), \, J^2(2), \, J^3(3), \, \ldots\big\}$.
\bigskip
{\bf 7.2 Theorem (General estimates, weakly convergent subsequences,
and Monge-Kantorovich H\"older continuity of heat distributions).}
{\sl
\smallskip\noindent
(1) For each positive integer $j$ the following is true.
$$\| Q_j(t)\|_{{\bf L}_2}
+\| 1/ Q_j(t)\|_{{\bf L}_2}
+\| T_j(t)\|_{{\bf L}_2}
+ \| U_j(t)\|_{{\bf L}_2} \le C_1 \leqno{(1.1)}$$
for each time $t \in \R^+$; here
$$C_1= \sup\left\{ c_{{}_\ell}^2, c_{{}_s}^{-2}\right\} \left(
C_\ast |\Omega| + c_{{}_s}^{-1}\EE_0 \right).$$
\smallskip
$$\int_{t \in (0, \, \infty)} \dir\big(T_j(t) \big) 
+ \dir \big(U_j(t)\big) \,
d {\cal L}^1 t \le C_2; \leqno{(1.2)} $$
here
$$C_2 = \kappa_\ast^{-1} \sup \left\{ c_\ast^{-1}, \, C_\ast\right\}
\EE_0.  $$
\smallskip\noindent
(1.3) For each $0 < M < \infty$,
$${\cal L}^1 \bigg( \R^+ \cap \left\{ t \colon 
\dir\big(T_j(t) \big) + \dir \big(U_j(t)\big) 
 > M \right\} \bigg) < C_1 / M.$$
\smallskip
\noindent
(2) There exists a subsequence $J_1$ of $J_0$ and a limit time
parametrized family of heat distributions $Q_\ast(t)$ in $\QQ$
such that
$$\left\|Q_{J_1(i)}(t) - Q_\ast(t) \right\|_\ast \to
0 \leqno{(2.1)}$$
locally uniformly in $t$ as $i \to \infty$.
$$\left\| Q_\ast(s) - Q_\ast(t) \right\|_\ast \le \rho
|s-t|^{1 \over 2}\leqno{(2.2)}$$
whenever $0 \le s < t < \infty$; here
$$\rho = { \kappa_0 |\O|^{1 \over 2} \over \kappa_\ast^{1
\over 2} c_\ast^{1 \over 2}} \EE_0.$$
\smallskip
\noindent
(3)  There is a subsequence $J_2$ of $J_1$ above
for which the  following is true.
\smallskip\noindent
(3.1)  There is a function $K_0$ in $\L_{1, loc}\big( \Omega
\times \R^+, \, [0, \, 1]\big)$ such that
$$\int_{(x, \, t) \in \Omega \times R^+}
\varphi(x, \, t) \left( \upchi_{K_{J_2(i)}(t)} (x) - K_0(x, \,
t) \right) \, d \LLL(x, \, t) \to 0$$
as $ i \to \infty$
for each $\varphi \in {\cal D}(\Omega \times \R)$; here
$$\upchi_{K_{J_2(i)}(t)}(x) = \cases{1 & if $ x \in K_{J_2(i)}(x)$\cr
			           $\phantom{x}$\cr
                                   0 & otherwise\cr}.$$
\smallskip\noindent
(3.2)  There are functions $Q_\ast$ (necessarily the same as
in (2) above) and $T_\ast$ in 
$\L_{2, loc}(\O \times R)$ such that, for each $0 < t <
\infty$,
$Q_\ast(t) \in \QQ$ and
$$\| Q_\ast(t)\|_{{\bf L}_2} +\| T_\ast(t) \|_{{\bf L}_2} \le C_1$$
and  that
$$\int_{\Omega \times R^+} \varphi \left( Q_{J_2(i)} - Q_\ast) \right)
\, d \LLL \to 0,$$
$$\int_{\Omega \times R^+} \varphi \left( T_{J_2(i)} - T_\ast) \right)
\, d \LLL \to 0$$
as $ i \to \infty$
for each $\varphi \in {\cal D}(\Omega \times \R)$.
For each fixed time $t$ corresponding to which there is a crystal
$\l K_\ast(t)\r$ in $\KK$ such that
$$\LL\big( K _{J_2(i)} \tt K_\ast(t) \big) \to 0 \quad \hbox{and}\quad
\big\|Q_{J_2(i)}(t) - Q_\ast(t) \big\|_{{\bf L}_2} \to 0$$
as $i \to \infty$, then we can (and do) require that 
$$c_{K_\ast(t)} T\ast(t) = Q_\ast(t).$$
\smallskip\noindent
(2.3)  There is a function $\nabla T_\ast$ in
$\L_{2}(\O \times \R^+, \, \RR)$ such that
$$\int_{\O \times(0, \, \infty)} \|\nabla T_\ast\|^2 \,
d \LLL \le C_2;$$
and  that
$$\int_{ \Omega \times R^+}
\varphi\left(\nabla T_{J_2(i)} - \nabla T_\ast\right) \, d \LLL \to 0$$
as $ i \to \infty$ and also
for each $\varphi \in {\cal D}(\Omega \times \R)$
$$\int_{ \Omega \times (0, \, \infty)}
\psi\nabla T_\ast - \nabla\psi T_\ast \, d \LLL = 0$$
for each $\psi \in {\cal D}\left(\Omega \times (0, \, \infty)\right)$
}
\bigskip
PROOF.
Assertion (1.1) follows from A.1.9 together with 5.3(1).
Assertion (1.2) follows from 5.2(2) and 5.3(4).  Assertion
(1.2) implies assertion (1.3).

 We use (1.1) together with the compactness estimate
in B.6 applied to the  heat distributions $Q_j(t)$ to
infer that, for each fixed time $t$ in $\R^+$, there exists a
subsequence $J$ of $J_0$ and
a limit heat distribution $Q_\ast(t)$ such that
$\left\| Q_{J(i)} - Q_\ast(t) \right\|_\ast \to 0$ as $i \to
\infty$.  Cantor's diagonal process guarantees the existence
of a further subsequence $J_1$ of $J_0$  such that
 such convergence occurs for
a countable set of $t$'s which is dense in $\R^+$.  The
estimate 5.3(2) implies the asserted convergence and H\"older
continuity as in Ascoli's theorem.

Conclusion (3) follows from conclusion (1), 2.5.5, and  the
usual weak compactness of closed and bounded subsets of vector
spaces of square summable functions as in B.3. \E
\bigskip
{\bf 7.3 Terminology.}
We fix a subsequence $J_2$ of $J_0$ for which conclusions (2) and (3)
of Theorem~7.2 both hold and let $K_0$, $Q_\ast$, $T_\ast$,
$\nabla T_\ast$ have the meaning of conclusions~(2) and (3) of 
Theorem~7.2.
\bigskip
{\bf 7.4 Theorem (Sufficient condition that $\l K_\ast(t)\r$,
$Q_\ast(x, \, t)$, $T_\ast(x, \, t)$ be a solution to
the heat equation with varying crystals).}
{\sl
If there is a subsequence $J$ of $J_2$ and a function
$\l K_\ast \r \colon \R^+  \to  \KK$ such that
$$\int_{t \in (0, \, c)} \LL\left(K_\ast(t) \tt
K_{J(i)}(t)\right)\, d {\cal L}^1 t \to 0$$
as $i \to \infty$ for each $0 < c < \infty$, then
the crystals $\l K_\ast(t)\r$ and functions
$Q_\ast(x, \, t)$, and $T_\ast(x, \, t)$ are a solution
to the heat equation with varying crystals with initial head
distribution $Q_0$ and final heat distribution $Q_\ast(\cdot,
\, b)$ for each $0 < b< \infty$.
}
\bigskip
PROOF.
With regard to condition 2.5.13, we let $\psi$ be a smooth
function which $\L_2$ closely approximates $c_{K_\ast}$ and then
write
$$
\eqalign{\varphi \left( c_{K_\ast} T_\ast - Q_\ast \right)
&= 
\varphi \left( c_{K_{J(i)}} T_{J(i)} - Q_{J(i)} \right)
+\varphi \left(  Q_{J(i)} - Q_\ast \right)\cr
&\phantom{XX}+\varphi \psi\left( T_\ast - T_{J(i)} \right)
+\varphi \left(c_{K_\ast} - \psi\right)
\left( T_\ast - T_{J(i)} \right)\cr
& \phantom{XX}+\varphi \left(c_{K_\ast} - c_{K_{J(i)}}\right)
T_{J(i)}\cr.}$$
It follows from 2.5.6 and 5.1 that
$$\int\varphi \left( c_{K_{J(i)}} T_{J(i)} - Q_{J(i)} \right)
\, d \LLL = 0.$$
It follows from 7.2(3.2) that
$$\int \varphi \left(  Q_{J(i)} - Q_\ast \right) \, d \LLL\to 0$$
and
$$\varphi \psi\left( T_\ast - T_{J(i)} \right) \, d \LLL \to
0$$
as $i \to \infty$.

We use Schwarz's inequality and 7.2(1.1)(3.2) to estimate
$$\left| \int \varphi \left(c_{K_\ast} - \psi\right)
\left( T_\ast - T_{J(i)} \right) \, d \LLL \right|
\le 2 b^{1 \over 2} C_1
\sup |\varphi | \left(\int \left(c_{K_\ast} - \psi\right)^2 \,
d \LLL 
\right)^{1 \over 2}$$
and
$$\left| \int\varphi \left(c_{K_\ast} - c_{K_{J(i)}}\right)
T_{J(i)} \, d \LLL\right|
\le 2 b^{1 \over 2}\sup |\varphi| c_{{}_\ell} \left( \int
\LL\left(K_\ast (t) \tt K_{J(i)}(t) \right) \, d {\cal L}^1
t \right)^{1 \over 2}.$$
The first integral is small if $\psi$ closely approximates
$c_{K_\ast}$ in $\L_2$ while the  second integral approaches zero as
$i \to \infty$ by assumption.

With regard to 2.5.14 we set $\Delta t = 2^{-j}$ and compare
$$ \int_{\O \times (0, \, b)}  \dotf Q_\ast + \kappa_{K_\ast} (
\nabla \varphi) \bullet \nabla T_\ast \, d \LLL
+ \int_\O \varphi(\cdot, \, b) Q_\ast(\cdot,\, b) -
\varphi(\cdot, \, 0) Q_0 \, d \LL$$
with
$$\int{\O \times (0, \, b)}  \dotf Q_j + \kappa_{K_j} (
\nabla \varphi) \bullet \nabla T_j \, d \LLL\phantom{XXXXXX}$$
$$\phantom{XXXXXX}
+ \sum_{k=1}^{b/\Delta t}
\int_\O \varphi(\cdot, \,k \Delta t) P_j(\cdot,\, k\Delta t) -
\varphi(\cdot, \, (k-1)\Delta t) Q_j(\cdot, \, (k-1)\Delta t) 
\, d \LL;$$
here for notational  simplicity $b$ is assumed to be an
integral multiple of $\Delta t$.  The second expression vanishes 
in accordance with 2.5.7 since in our approximations
the crystal is fixed in each $\Delta t$ interval.
It is a minor extension of
the present argument to treat the general case.

\noindent
An argument similar to that used in checking 2.4.13 shows
that
$$ \int_{\O \times (0, \, b)}  \dotf Q_{J_1(i)} + \kappa_{K_{J_1(i)}}
( \nabla \varphi) \bullet \nabla T_{J_1(i)}j \, d \LLL \phantom{XXX}$$
$$\phantom{XXXXXXX} \to
\int_{\O \times (0, \, b)}  \dotf Q_\ast + \kappa_{K_\ast} (
\nabla \varphi) \bullet \nabla T_\ast \, d \LLL$$
as $i \to \infty$.

\noindent
For the $\O$ integrals we estimate
$$\eqalign{ \sum_{k=1}^{b/\Delta t}
&\int_\O \varphi(\cdot, \,k \Delta t) P_j(\cdot,\, k\Delta t) -
\varphi(\cdot, \, (k-1)\Delta t)
 Q_j(\cdot, \, (k-1)\Delta t) \, d \LL\cr
&\phantom{XXX}- \int_\O \varphi(\cdot, \, b) Q_\ast(\cdot,\, b) -
\varphi(\cdot, \, 0) Q_0 \, d \LL \cr
& =\sum_{k=1}^{b/\Delta t- 1}
\int_\O \varphi(\cdot, \,k \Delta t)\left( P_j(\cdot,\, k\Delta t) 
-Q_j(\cdot, \, k\Delta t)\right) \, d \LL \cr
&\le \sum_{k=1}^{b/\Delta t- 1}
\lip(\varphi) \left\| P_j(\cdot,\, k\Delta t) -
-Q_j(\cdot, \, k\Delta t)\right\|_\ast\cr
&\le \lip(\varphi) \sum_{k=1}^{b/\Delta t- 1}
\Delta t^{-j\alpha} \left( \EE_j((k \Delta t)^-) - \EE_j(k
\Delta t)\right)\cr
&\le \Delta t^{-j\alpha} \,\lip (\varphi) \, \EE_0.\cr}$$
Hence we have limit zero as $j \to \infty$.  \E
\bigskip
{\bf 7.5 Terminology.}  

(1) We denote by $Y$ the collection of all
triples $(\l K \r, \, Q, \, T)$ for which $\l K \r$ is a
crystal in $\KK$, $Q$ is a heat distribution in $\QQ$, and 
$T = c_K^{-1}Q$ is the associated temperature field.  We equip
$Y$ with a metric
$$\mu \colon Y \times Y \to \R^+,$$
$$\eqalign{\mu\bigg(
\big(\l K \r, \, P, \, S\big), \big(\l L \r, \, Q, \, T\big)\bigg)
&= \LL(K \tt L) \cr
&\phantom{XXX}+ \| P-Q \|_{{\bf L}_2}
+ \| 1/ P - 1/ Q \|_{{\bf L}_2} \cr
&\phantom{XXXXX}+ \| S-T \|_{{\bf L}_2}
+ \| 1/ S - 1/ T \|_{{\bf L}_2}.\cr}$$
\smallskip
(2) For each $0 < M < \infty$ we denote
$$Y(M) = Y \cap \big\{ 
(\l K \r, \, Q, \, T)
\colon \dir (T) + \dir (1/T) \le M \big\}.$$
\bigskip
{\bf 7.6 Theorem (Compactness of the $Y(M)$'s and the modulus of
continuity functions $\sigma_M$).}
\smallskip\noindent
{
\sl
(1) Corresponding to each $0 < M < \infty$ and each sequence
$\big(\l K^i \r, \, Q^i, \, T^i\big) \in Y(M)$, $i = 1, \,2, \, 3, \, \ldots$
there exists a subsequence $J$ of $J_0$ together with
$\big(\l K \r, \, Q, \, T\big) \in Y(M)$ such that, as $i \to
\infty$,
$$\mu\bigg( \big(\l K^{J(i)} \r, \, Q^{J(i)}, \, T^{J(i)}\big), \,
\big(\l K \r, \, Q, \, T\big)\bigg) \to 0 \quad \hbox{and} \quad
\| Q^{J(i)} - Q \|_\ast \to 0$$
and also
$$\nabla T^{J(i)} \h \nabla T \quad\hbox{and}\quad
\nabla\left(1/T^{J(i)}\right) \h \nabla\left(1/T\right)\quad\big(
\hbox{weak convergence in } \L_2\left(\O, \, \R^n\right)\big).$$
\smallskip\noindent
(2)  Corresponding to each $0 < M < \infty$ there is an increasing
function $\sigma_M \colon \R^+ \to \R^+$ with
$\lim_{r \downarrow 0} \sigma_M(r) = 0$ such that
$$ \mu\bigg( \big(\l K \r, \, P, \, S\big), \,
\big(\l L \r, \, Q, \, T\big)\bigg)
 \le \sigma_M\big( \| P - Q \|_\ast \big)$$
whenever
$(\l K \r, \, P, \, S), \, (\l L \r, \, Q, \, T)$ belong to $Y(M)$.
}
\bigskip
PROOF
The compactness theorem for integral currents implies the convergence
of a subsequence of the $\l K^i\r$'s.  Rellich's theorem (B.7)
implies the asserted convergences for a further subsequence of the
$T^i$'s, $1/T^i$'s, $\nabla T^i)$'s, and $\nabla\left(1/
T^i\right)$'s.  The convergence of the
$Q^i$'s and $1/Q^i$'s follows, which gives the first part of the 
first assertion.
The second part of the first assertion follows from the first.
With regard to the second assertion, we see from Corollary~5.5
that
 $\LL(K \tt L)$
is small if
$\| P - Q \|_\ast$ 
is small, and from Proposition~6.3 that
$\| S - T \|_{{\bf L}_2}  +\| 1/S - 1/T \|_{{\bf L}_2}$
is small if
$\LL(K \tt L)$ and $\| P - Q \|_\ast$ are both small.  As above,
these conditions imply that
$\| P - Q \|_{{\bf L}_2}  +\| 1/P - 1/Q \|_{{\bf L}_2}$
is small.  The second assertion follows.  \E
\bigskip
{\bf 7.7 Main Convergence Theorem.}
{\sl
There is a subsequence $J$ of $J_2$ together with varying
crystals $\l K_\ast(t) \r$ in $\KK$, varying heat distributions
$Q_\ast(t)$ in $\QQ$, and associated varying temperature
fields $T_\ast(t) =  c_{K_\ast(t)}^{-1}Q_\ast(t)$ for which the
following is true.
\smallskip\noindent
(1) For ${\cal L}^1$ almost every time $t$ in $\R^+$
\smallskip\noindent
$$ \mu\bigg( \big(\l K_{J(i)}(t) \r, \, Q_{J(i)}(t), 
\, T_{J(i)}(t)\big), \,
\big(\l K_\ast(t) \r, \, Q_\ast(t), \, T_\ast(t)\big)\bigg) \to 0$$
%and also
%$$\nabla T_{J(i)}(t) \h \nabla T_\ast (t)\quad\hbox{and}\quad
%\nabla\left(1/T_{J(i)}(t)\right) \h \nabla\left(1/T_\ast(t)\right)$$
%(weak convergence in $\L_2\left(\O, \, \R^n\right)$) as $i \to \infty$
with
$$\lim \inf_{i \to \infty}\dir\big(T_{J(i)}(t)\big) <
\infty\quad\hbox{and}\quad
\lim \inf_{i \to \infty}\dir\big(1/ T_{J(i)}(t)\big) <
\infty.$$
\smallskip\noindent
(2)  The crystals $\l K_\ast(t)\r$ and functions $Q_\ast(x, \, t)$
and $T(x, \, t)$ are a solution to the heat equation with varying
crystals as in 2.5 with initial crystal $\l K_0 \r$ and initial
heat distribution~$Q_0$.
\smallskip\noindent
(3)  The union
$$\bigcup\big\{ K_{\ast}(t)\colon 0 \le t < \infty\big\}$$
of the crystal sets $K_\ast(t)$
is $\LLL$ measurable.
}
\bigskip
PROOF.
\smallskip
{\bf Step 1.}  We choose and fix a sequence of intervals
$I_1, \, I_2, \, I_3, \, \ldots$ in $\R^+$ with diameters converging
to zero such that $\bigcup_{\ell = k}^\infty I_\ell = \R^+$ for each
positive integer~$k$.
\bigskip
{\bf Step 2.}  Suppose $M$ is a positive integer and $J$ is any 
subsequence of $J_0$ and $S$ is the family of times $t$ such there are
infinitely many $j$ in $\left\{ J(1), \, J(2), \, J(3), \, \ldots
\right\}$ such that
$$\dir\big( T_j(t)\big)
+\dir\big( U_j(t)\big) \le M.$$
Then ${\cal L}^1 \left( \R^+ \sim S\right) \le C_1/M$ where $C_1$ is 
the constant of 7.2(1.3).
\smallskip
{\bf Proof.}
Let $X(k)$ denote the set of times $t$ such that whenever $i \ge k$
then
$$\dir\big( T_{J(i)}(t)\big)
+\dir\big( U_{J(i)}(t)\big) > M$$
so that $X(1) \subset X(2) \subset X(3) \subset \ldots$.  Since
$X(k)$ is a subset of the $t$'s for which
$$\dir\big( T_{J(k)}(t)\big) +\dir\big( U_{J(k)}(t)\big) > M$$
we infer from 7.2(1.3) that ${\cal L}^1(X(k)) \le C_1/M$.
The assertions follows since
$$\R^+ \sim S = \cup \big\{ X(k) \colon k = 1, \, 2, \, 3, \,
\ldots\big\}.$$
\bigskip
{\bf Step 3.}
Our proof requires that we construct for all positive integers $M$ 
and $k$ the following:
\smallskip
\noindent
{\bf (a)} Subsequences $J^M$, $J^M_\ast$, $M^{M,k}$
of $J_0$ subject to the following conditions:
\smallskip
(a.1)  $J^{1,1} = J_2$ as in 7.5.
\smallskip
(a.2)  $J^M$ result of Cantor's diagonal process applied
to the sequences $J^{M,1}$, $J^{M,2}$, $J^{M,3}, \, \ldots$.
\smallskip
(a.3) $J^M_\ast$ is a subsequence of $J^M$ selected in accordance 
with Step~4 below.
\smallskip
(a.4) $J^{M+1,1} = J^M_\ast$.
\smallskip
Our final sequence $J$ is the result of Cantor's diagonal process
applied to the sequences
$J^M_\ast$, $J^{M+1}_\ast$, $J^{M+2}_\ast, \, \ldots$.
\medskip
\noindent
{\bf (b)} Sets of times $S(M)$, $S_\ast(M)$, $S\left(M, \, J^{M,k}\right)$
subject to the following conditions:
\smallskip
(b.1) $S\left(M, \, J^{M,k}\right)$ is the set of times $t$ in $\R^+$
for which there are infinitely many $j$'s in $\left\{
J^{M,k}(1), \, J^{M,k}(2), \, J^{M,k}(3), \, \ldots \right\}$
for which
$$\dir \big(T_j(t)\big) + \dir\big( U_j(t) \big) \le M.$$
We infer that $S\left(M, \, J^{M,k+1}\right)\subset
 S\left(M, \, J^{M,k}\right)$.  We further infer from Step~2
that 
$${\cal L}^1\big( \R^+ \sim S\left(M, \, J^{M,k}\right)\big) \le C_1/M.$$
\smallskip
(b.2)  $S(M) = \bigcap \left\{ S\left(M, \, J^{M,\ell}\right) \colon
\ell = 1, \, 2, \, 3, \, \ldots \right\}$.  Since
$$\R^+ \sim S(M) = \bigcup \left\{ \R^+ \sim
 S\left(M, \, J^{M,\ell}\right) \colon
\ell = 1, \, 2, \, 3, \, \ldots \right\},$$
 we infer
${\cal L}^1\left(\R^+ \sim S(M)\right) \le C_1/M$.
\medskip
(b.3)  $S_\ast(M) \subset \R^+$ with
${\cal L}^1\left(S(M) \sim S_\ast(M)\right) =0$ is constructed below.
\smallskip
\noindent
{\bf (c)} Countable families $W(M) = \left\{ t^{M,1}, \, t^{M,2}, \,
t^{M,3}, \, \ldots\right\}$ of times subject to the following
constraints:
\smallskip
(c.1)  $t^{M,k} \in
 S\left( M, \, J^{M,k}\right)$.
\smallskip
(c.2) If $I_k \cap 
S\left( M, \, J^{M,k}\right) \not= \emptyset$, then
$t^{M,k}$ also lies in the interval~$I_k$,
\smallskip
(c.3)  If $M \ge 2$ and it is consistent with the requirements (c.1) 
and (c.2) above then $t^{M,k} = t^{M-1,k}$.
\smallskip
It follows from our construction and constraint (b.1) and the 
choice of $I_k$'s in Step~1 that
\smallskip
(c.4)  $W(M)\subset S(M)$, and $W(M)$ is dense in $S(M)$.
\medskip
Our construction procedure is the following:
\smallskip
\noindent
{\bf (1)}  Assuming $J^{M,k}$ has been constructed we pick
$t^{M,k} \in S\left(M, \, J^{M,k}\right)$ in accordance with
(c) above.
\smallskip
\noindent
{\bf (2)}  Assuming $J^{M,k}$ and $t^{M,k}$ have been fixed, we use
conclusion (1) of Theorem~7.6 to choose a subsequence $J^{M,k+1}$
of $J^M$ together with a limit
$$\big(\l K_\ast(t^{M,k}) \r, \, Q_\ast(t^{M,k}), \, T_\ast(t^{M,k}) 
\big) \in Y(M)$$
such that, abbreviating $j(i) = J^{M,k+1}(i)$, we have
$$\mu\bigg(
\big(\l K_{j(i)}(t^{M,k}) \r, \, Q_{j(i)}(t^{M,k}), \,
 T_{j(i)}(t^{M,k}) \big),\,
\big(\l K_\ast(t^{M,k}) \r, \, Q_\ast(t^{M,k}), \, T_\ast(t^{M,k}) 
\big)\bigg) \to 0$$
and also
$$\nabla T_{j(i)}(t^{M,k}) \h \nabla T_\ast(t^{M,k})
\quad\hbox{and}\quad
\nabla\left(1/T_{J(j)}(t^{M,k})\right) \h \nabla\left
(1/T_\ast(t^{M,k})\right)$$
(weak convergence in $\L_2\left(\O, \, \R^n\right)$)
as $i \to \infty$.
\smallskip
\noindent
{\bf (3)} It follows by construction that
$Q_\ast(t^{M,k})$ and  $T_\ast(t^{M,k})$ are the heat distribution 
and temperature field specified in 7.3; note 7.2(2)(3).
\smallskip
\noindent
{\bf (4)}  It further follows from 7.2(2.2) and 7.6(2) that whenever
$s, \, t \in W(M)$ then
$$\mu\bigg(
\big(\l K_\ast(s) \r, \, Q_\ast(s), \, T_\ast(s) \big), \,
\big(\l K_\ast(t) \r, \, Q_\ast(t), \, T_\ast(t) \big)\bigg)$$
$$\phantom{XXXXXXXXXX} \le \sigma_M\left(\|Q(s) - Q(t)\|_\ast \right)
\le \sigma_M\left(\rho |s-t|^{1 \over 2}\right).$$
Since $Y(M)$ is complete there is a unique continuous extension
of the function
$$\big(\l K_\ast \r, \, Q_\ast, \, T_\ast \big) \colon W(M) \to Y(M)$$
to a function
$$\big(\l K \r, \, Q, \, T \big) \colon \hbox{Closure}\big[W(M)\big]
\to Y(M).$$
\smallskip
We do not claim convergence of the
$$\big(\l K_{J^M(i)}(t) \r, \, Q_{J^M(i)}(t), \,
 T_{J^M(i)}(t) \big)\hbox{'s}$$
for all $t$'s in Closure$\big[W(M)\big]$.
The construction of $J_\ast(M)$ and $S_\ast(M)$ for which there
is convergence is given in Step~4 below.
\bigskip
{\bf Step 4.}
We assert convergence in measure with respect to  the $\mu$
metric of the
$\big(\l K_{J^M(i)}(t) \r, \, Q_{J^M(i)}(t), \,
 T_{J^M(i)}(t) \big)$'s to
the $\big(\l K(t) \r, \, Q(t), \, T(t)\big)$'s
as $i \to \infty$ for $t$'s in Closure$\big[W(M)\big]$.
As a consequence of this convergence in measure there is a subsequence
$J^M_\ast$ of $J^M$ (as in Step~3(a.4)
above) and an ${\cal L}^1$ measurable set $S_\ast(M)
\subset \hbox{Closure}W(M)$ (as in Step~3(b.3) above) with
$${\cal L}^1\bigg( \hbox{Closure}\big[W(M)\big] \sim S_\ast(M)\bigg)
={\cal L}^1\big( S(M) \sim S_\ast(M)\big) = 0$$
such that, for each $t \in S_\ast(M)$,
$$\mu\bigg(\big(\l K_{J^M(i)}(t) \r, \, Q_{J^M(i)}(t), \,
 T_{J^M(i)}(t) \big), \,
\big(\l K(t) \r, \, Q(t), \, T(t) \big)\bigg) \to 0$$
as $i \to \infty$.  For such $t$'s we infer $Q(t) = Q_\ast(t)$,
$T(t) = T_\ast(t)$, and we set $\l K_\ast(t)\r = \l K(t) \r$.
\smallskip
{\bf Proof.}
We will verify the asserted convergence in  measure.
Suppose then $\epsilon > 0$ and $0 < b < \infty$.  We will show
that, for all sufficiently large $j$'s in $\left\{J^M(1), \,
 J^M(2), \, J^M(3), \, \ldots\right\}$,
the one dimensional Lesbegue measure of the set
$$ [0, \, b] \cap \hbox{Closure}\big[W(M\big]) \cap
\bigg\{ t \colon
\mu \bigg(\big(\l K_j(t) \r, \, Q_j(t), \,
 T_j(t) \big), \, 
\big(\l K(t) \r, \, Q(t), \, T(t) \big)\bigg) > \epsilon \bigg\} $$
is less than $\epsilon$.
There are five steps in the demonstration.
\smallskip
\noindent
{\bf (1)}
First we use 7.2(1.3) to pick and fix an integer $N \ge M$
sufficiently large so that, for each $j \in \{ J^M(1), \, J^M(2), \,
J^M(3), \, \ldots\}$, if
$$W_j = \bigg\{ t \colon \dir \big(T_j(t)\big)
+ \dir\big(U_j(t)\big) \le N \bigg\}$$
then ${\cal L}^1 \big( [0, \, b] \sim W_j\big) < \epsilon$.
\smallskip
\noindent
{\bf (2)}
We now use 7.6(2) to pick and fix $\Delta Q > 0$ sufficiently
small so that $\sigma_N(4\Delta Q) < \epsilon$.
\smallskip
\noindent
{\bf (3)}
We fix a finite set of times $t_1, \,t_2, \, \ldots, \, t_m$
in W(M) such that, for each $t$ belonging to $[0, \, b] \cap 
\hbox{Closure}W(M)$, there is $\alpha \in \{1, \, 2, \, \ldots, \, 
m\}$ such that
$$\|Q(t) - Q(t_\alpha)\|_\ast < \Delta Q\quad \hbox{and} \quad
\rho\left| t - t_\alpha\right| < \Delta Q.$$
\smallskip
\noindent
{\bf (4)}
We infer from 5.3(2), 7.2(2.1), Step~3(a.1), and (3) above that
for all sufficiently large $j$ in $\left\{J^M(1), \, J^M(2), \,
J^M(3), \, \ldots\right\}$ it will be true for each
$t$ belonging to  $[0, \, b] \cap \hbox{Closure}\big[W(M)\big]$,
 that there is
$\alpha \in \{1, \, 2, \, \ldots, \, m\}$ such that
$$\|Q_j(t) - Q_j(t_\alpha)\|_\ast < 2 \Delta Q\quad, \quad
\|Q_j(t_\alpha) - Q(t_\alpha)\|_\ast < \Delta Q, \quad \hbox{and} \quad
\|Q(t) - Q(t_\alpha)\|_\ast < \Delta Q$$
so that
$$\|Q_j(t) - Q(t)\|_\ast < 4\Delta Q$$
\smallskip
\noindent
{\bf (5)}
For each fixed sufficiently large $j$ in $\left\{J^M(1),
\, J^M(2), \, J^M(3), \, \ldots\right\}$ we use (1),
(2), (3), (4) to estimate
$${\cal L}^1 \bigg([0, \, b] \cap \hbox{Closure}\big[W(M)\big] 
\sim W_j\bigg) < \epsilon$$
while, for each $t \in
 \bigg([0, \, b] \cap \hbox{Closure}\big[W(M)\big] \sim W_j\bigg)$,  
$$\big(\l K_j(t)\r, \, Q_j(t) , \, T_j(t)\big)\quad\hbox{and}\quad
\big(\l K(t)\r, \, Q(t) , \, T(t)\big)$$
both belong to $Y(M)$ with
$$\sigma_N\big(\|Q_j(t) - Q_(t) \|_\ast\big) 
< \sigma_N(4 \Delta Q) < \epsilon.$$
\bigskip
{\bf Step 5.}  Conclusion (1) of our theorem now follows from the
construction in view of Step~2(b.2) and Steps~4 and 5.  Our 
subsequence of subsequence construction guarantees that convergence is
never lost for any time for which it is established.  Conclusion~(2) of
our theorem follows from conclusion~(1) and Theorem~7.4.
By construction in 2.11, the characteristic function
$\upchi_{J(i)}$
of the set
$\bigcup\big\{ K_{J(i)}(t)\colon 0 \le t < \infty\big\}$
is $\LLL$ measurable for each~$i$.  We infer from (b.2),
(b.3), (c.4) and Step~4 above that the $\upchi_{J(i)}$'s converge
in measure to the characteristic function $\upchi_\ast$ of the set
$\bigcup\big\{ K_{\ast}(t)\colon 0 \le t < \infty\big\}$.
We infer that $\upchi_\ast$ is $\LLL$ measurable so that
$\bigcup\big\{ K_{\ast}(t)\colon 0 \le t < \infty\big\}$
is also $\LLL$ measurable.  This is Conclusion~(3) of our theorem.
\E
\bigskip
\centerline{\larger 8. The Gibbs-Thomson condition holds for
elliptic $\Phi$'s}
\bigskip
According to Theorem~4.4 the Gibbs-Thomson relation between
first variation distributions and temperature holds exactly
for the $\EEE$ minimizers constructed in our approximating
evolutions in case our surface energy function $\Phi$ is
smooth.  In the present chapter we show that in case $\Phi$ is
elliptic then the Gibbs-Thomson relation also holds in the
final evolution which is  a limit of the approximating ones.
The sequence of approximations which leads this equality
is the following.
First we fix a time $t$ which is ``good'' according to the
criteria of Theorem~8.1 below (almost every time is good).
We then fix an initial velocity vector field $g_\ast$ and
associated deformation $G_s(x) = x + s\, g_\ast(x)$
Associated with each large integer $i$ is the time
$t_0 = t_0(i)$ at the the beginning of the $\Delta t =
2^{-J(i)}$ time interval
which contains $t$ in construction of the $J(i)$-th
approximating evolution.  For fixed $i$ we have:
$$\leqalignno{
{d \over ds}\Phi\Big(G_{s\sharp}
\partial \l K_\ast(t)\r\Big)
&
\Bigg|_{s=0}
\cr
&
\approx
{d \over ds}\Phi\Big(G_{s\sharp}\partial \l
K_{J(i)}(t)\r\Big)\Bigg|_{s=0}
\cr
&&\hbox{by Theorem 8.6(3) below}
\cr &
=
\int_{x \in \partial K_{J(i)}(t_0)}
H\left(T_{J(i)}(x, \,t_0)\right)\, g_\ast\bullet {\bf
n}_{K_{J(i)}(t_0)}
\, d \HH x
\cr 
&&\hbox{by Theorem 4.4}
\cr &
\approx
\int_{x \in \partial K_{J(i)}(t)}
H\left(T_{J(i)}(x, \,t)\right)\, g_\ast\bullet {\bf
n}_{K_{J(i)}(t)} \, d \HH x
\cr 
&&\hbox{by Theorem 8.3 below}
\cr &
=
\int_{x \in K_{J(i)}(t)}
\hbox{div}\left(H\left(T_{J(i)}(x, \,t)\right)\, g_\ast\right)
\, d \LL x
\cr 
&&\hbox{by the Gauss-Green theorem [FH 4.5.6(5)]}
\cr &
\approx
\int_{x \in K_\ast(t)}
\hbox{div}\left(H\left(T_\ast(x, \,t)\right)\, g_\ast\right)
\, d \LL x
\cr 
&&\hbox{by Theorem 8.1 below}
\cr &
= \lim_{\epsilon \downarrow 0}
\int_{x \in \partial K_\ast(t)} H_\epsilon \big(T_\ast(x, \, t)\big)
\, {\bf n}_{K_\ast(t)}(x)\bullet g_\ast(x)\, d \HH x
\cr 
&&\hbox{by Theorem 8.7 below}
\cr}$$
Equalities result in the limit as $i \to \infty$.
In Theorem~8.6 we show that $\partial \l K_\ast(t)\r$ itself
as a solution to a variational problem.  This enables us in
Theorem~8.8 to conclude $\partial K_\ast(t)$ is a smooth
one dimensional submanifold of $\O$ without boundary
in  case $n=2$ and is a
homeomorphic embedding of a compact continuous two 
dimensional
manifold without boundary in case $n=3$.
\bigskip
{\bf 8.1 Theorem (Some integral convergences for sequences
of crystals).}
{\sl
Suppose, for some fixed time $t$ there is a subsequence (just
denoted by the index $j$) of the integers for which
the following is true as $j \to \infty$.
\smallskip\noindent
(a) $\LL\big(K_j(t) \tt K(t)\big) \to 0$;
\smallskip\noindent
(b) $T_j(t) \to T(t)$;
\smallskip\noindent
(c) $U_j(t) \to U(t)$;
\smallskip\noindent
(d) $\nabla T_j(t) \h \nabla T(t)$ with $\sup_j\dir(T_j(t)) <
\infty$;
\smallskip\noindent
(e) $\nabla U_j(t) \h \nabla U(t)$ with $\sup_j\dir(U_j(t)) <
\infty$.
\smallskip\noindent
Then, for  each  smooth function $g \colon \O \to \R$ and for
each $i = 1, \, 2, \, \ldots, \, n$,
\smallskip\noindent
$$\int_{x \in K_j(t)} {\partial \over \partial x_i}\bigg(
H\big(T_j(x, \, t) \big)\, g(x)\bigg) \, d \LL x 
\to
\int_{x \in K(t)} {\partial \over \partial x_i}\bigg(
H\big(T(x, \, t) \big)\, g(x)\bigg) \, d \LL x$$
as $j \to \infty$.
}
\bigskip
PROOF.  It clearly suffices to consider only the case $i =
n$.
Also,  in view of [MC Lemma 3.1.1] we can assume without loss
of generality that the functions
$$x_n \mapsto T_j\big(x_1, \, \ldots, \,  x_{n-1}, \, x_n, \,
t\big), \quad
x_n \mapsto T\big(x_1, \, \ldots, \,  x_{n-1}, \, x_n, \,
t\big)$$
are simultaneously absolutely continuous for each $j$ for
${\cal L}^{n-1}$ almost all $\big(x_1, \, \ldots, \, x_{n-1}\big)$
(with the obvious meaning in $\O$).  We can therefore evaluate
$${\partial \over \partial x_i}\bigg(
H\big(T_j(x, \, t) \big)\, g(x)\bigg)
= H'\big(T_j(x,  \, t)\big)
{\partial T_j\over \partial x_i}(x, \, t) \,  g(x)
+ H\big(T_j(x, \, t) \big))
{\partial g \over \partial x_i}(x).$$
For notational simplicity we will drop references to the
variables $x$ and the
variable $t$ (since it is fixed) and will abbreviate by
$\upchi_j$ the characteristic function of $K_j$ for  each  $j$
and similarly 
will abbreviate by $\upchi$ the characteristic 
function of $K$.

We infer from the equations A.1.6 and A.1.6 the existence  of
bounded Lipschitz functions $J, \, L \colon \R^+ \to \R$ such
that
$$H(T_\ast) = 
\left( c_{{}_\ell} - c_{{}_s} \right)
\left(
3U_\ast^2 - T_\ast^2 + J(T_\ast)
\right)
, \quad
H'(T_\ast) = 
\left( c_{{}_\ell} - c_{{}_s} \right)
\left(
-6U_\ast^3 - 2T_\ast + L(T_\ast)
\right)$$
for each $0 < T_\ast < \infty$ with $U_\ast = 1/T_\ast$.
We note
$${\partial U_j \over \partial x_n} = -U_j^2 {\partial
T_j\over x_n}, \quad
{\partial U \over \partial x_n} = -U^2 {\partial T\over x_n}$$
for each  $j$ and estimate
$$\eqalign{
\upchi_j{\partial \over \partial x_n}\bigg(H\left(T_j\right)\,
& g \bigg)
-\upchi{\partial \over \partial x_n} \bigg( H\left(T\right)\,
g\bigg)
\cr
&=
\upchi_j H'\left(T_j\right) {\partial T_j \over \partial x_n} \, g
-\upchi H'\left(T\right) {\partial T \over \partial x_n} \, g
\cr
&\phantom{xxxxXXXXXXXxxxxx}
+\upchi_j H\left(T_j\right) {\partial g \over \partial x_n}
-\upchi H\left(T\right) {\partial g \over \partial x_n}
\cr
&=
g \left( c_{{}_\ell} - c_{{}_s} \right)
\Bigg(
\upchi_j\big[ -6 U_j^3 - 2T_j + L(T_j) \big] {\partial T_j
\over \partial x_n}
\cr
&\phantom{XXXXXXXXXXXXXX}
- \upchi\big[ -6 U^3 - 2T + L(T) \big] {\partial T \over \partial x_n}
\Bigg)
\cr
&\phantom{x}
+ {\partial g\over \partial x_n}
\left( c_{{}_\ell} - c_{{}_s} \right)
\Bigg(
\upchi_j\big[ 3 U_j^2 - 2T_j^2 + J(T_j) \big]
- \upchi\big[ 3 U^2 - 2T^2 + J(T) \big]
\Bigg)
\cr
&=
g \left( c_{{}_\ell} - c_{{}_s} \right)
\Bigg(6\left[
\upchi_j U_j{\partial U_j \over \partial x_n} -
\upchi U{\partial U \over \partial x_n}
\right]
-2\left[
\upchi_j T_j{\partial T_j \over \partial x_n} -
\upchi T{\partial T \over \partial x_n}
\right]
\cr
&\phantom{xxxxxxxxxxxxXXXXXXXXXX}
+\left[
\upchi_j L\left( T_j\right){\partial T_j \over \partial x_n}
- \upchi L\left( T \right){\partial T \over \partial x_n}
\right]
\Bigg)
\cr
&
\phantom{xxxxxxxxx}
+ {\partial g \over \partial x_n}
\left( c_{{}_\ell} - c_{{}_s} \right)
\Bigg(
3 
\left(\upchi_j - \upchi\right) U^2
+
3
\upchi_j \left(U_j - U\right) \left(U_j + U\right)
\cr
&\phantom{XXXXXXXXXXXXXX}
+ 
\left(\upchi_j - \upchi\right) T^2
+
\upchi_j \left(T_j - T\right) \left(T_j + T\right)
\cr
&\phantom{XXXXXXXXXXXXXX}
+ 
\left(\upchi_j - \upchi\right) J(T)
+
\upchi_j \left(J\left(T_j\right) - J\left(T\right)\right)
\Bigg)
\cr
}$$
We write
$$\upchi_jU_j - \upchi U = \left( \upchi_j - \upchi\right) U 
+ \upchi_j \left(U_j - U \right),\quad
\upchi_jT_j - \upchi T = \left( \upchi_j - \upchi\right) T 
+ \upchi_j \left(T_j - T \right),$$
$$ \upchi_jL\left(T_j\right) - \upchi L(T)
= \left( \upchi_j - \upchi\right)L(T) 
+ \upchi_j \left(L(T_j) - L(T) \right)$$
and infer
$$g
\left( c_{{}_\ell} - c_{{}_s} \right)
6\upchi_j U_j \to g
\left( c_{{}_\ell} - c_{{}_s} \right)
6\upchi U,$$
$$g
\left( c_{{}_\ell} - c_{{}_s} \right)
(-2)\upchi_j T_j \to g
\left( c_{{}_\ell} - c_{{}_s} \right)
(-2)\upchi T,$$
$$g
\left( c_{{}_\ell} - c_{{}_s} \right)
\upchi_jL\left(T_j\right)\to g
\left( c_{{}_\ell} - c_{{}_s} \right)
\upchi L(T)$$
as $j \to \infty$ (strong convergence in $\L_2(\O)$) while
$${\partial U_j \over \partial x_n} \h
{\partial U \over \partial x_n}, \quad
{\partial T_j \over \partial x_n} \h
{\partial T \over \partial x_n}$$
as $j \to \infty$ (weak convergence  in $\L_2(\O)$).
We can then invoke the convergence properties set forth in
Appendix B.3 to conclude that the $\LL \rightangle \O$
integrals of functions appearing in the first summand of the
final expression in our sequence of equalities above converge
to zero as $j \to \infty$.  We invoke the absolute continuity
of integrals of summable functions
set forth in [FH 2.4.11] to conclude that the
$\LL\rightangle \O$ integrals of the first, third, and fifth functions
appearing in the second summand in the same final expression
also converge to zero as $j \to \infty$.  Finally we use
Schwarz's inequality in checking that the
$\LL\rightangle \O$ integrals of the second, fourth, and sixth functions
appearing in the same second summand of the same final expression
also converge to zero as $j \to \infty$.
This completes the proof.  \E
\bigskip
{\bf 8.2 Theorem (Some oriented integral estimates over
boundaries of crystals).}
{\sl
Suppose $K$ is a crystal in $\KK$
and $f \colon \O \to \R$ is a Lipschitz function having
support in an open ball $\,\U^n\big(p, \, \rho_\ast/2\big)$ in $\O$.
Then, for each positive integer $M$, the oriented surface integral
$$ \Bigg|\int_{p \in \partial K} f(p)\, {\bf n}_K(p) \bullet {\bf
e}_n\, d \HH p\,\Bigg|$$
does not exceed the sum
$$2^{3 \over 2}
M \bigg( n \alpha(n)\left({\rho_\ast \over 2}\right)^{n-1}
\bigg)^{1 \over 2}
\left(
\int_\O f^2 \, d \LL
\right)^{1 \over 4}
\left(
\int_\O |\nabla f |^2 \, d \LL
\right)^{1\over 4} 
\phantom{XXXXXX} $$
$$\phantom{XXXXXXXXXXXXXX}
+ \left(
{\rho_\ast\big(
\HH (\partial K) + n \alpha(n)\left({\rho_\ast \over 2}\right)^{n-1}
\big)
\over 2M}
\right)^{1 \over 2}
\left(
\int_\O |\nabla f|^2\,d \LL
\right)^{1 \over 2}.$$
If we replace $2^{3\over 2}$ by $2^{3\over 2} + 1$ in the above
sum, then the inequality holds for all $M\ge2$ (not just
integers). 
}
\bigskip
PROOF.
\smallskip
{\bf Step 1.} If $f \colon \R \to \R^+$ is a Lipschitz
function having compact support with
$$h = \sup f
\quad \hbox{and}\quad 2r = {\cal L}^1\{ s \colon f(s) \ge h/2\},$$
then
$$r\, h^2 \le 2 \int_{\bf R} f^2 \, d {\cal L}^1 \quad\hbox{and} \quad
{h^2 \over r} \le 2 \int_{\bf R} \left(\partial f \over \partial
r\right)^2\, d {\cal L}^1.$$
\smallskip
{\bf Proof.}  If $g \colon \R\to \R^+$ is the symmetric
decreasing rearrangement [AL]  of $f$ then
$$\sup g = \sup f = h, \quad \int_{\bf R} g^2 = \int_{\bf R} f^2,
\quad\hbox{and }
\quad \int_{\bf R} \left({\partial g \over \partial r}\right)^2\,
d {\cal L}^1
\le  \int_{\bf R}
\left({\partial f \over \partial r}\right)^2\, d {\cal L}^1.$$
Hence it is sufficient to prove our assertion in case $g = f$
which is left to the reader.
\bigskip
{\bf Step 2.} Suppose $f \colon \R^n \to \R^+$ is a Lipschitz
function having compact support and functions
$h, \, r \colon \R^{n-1} \to \R^+$ are defined by setting
$$h(x) = \sup \{f(x, \, s) \colon s \in \R\},
\quad 2r(x) = {\cal L}^1\{ s \colon f(x, \, s) \ge h(x)/2\}$$
for each $x \in \R^{n-1}$.
Then, for each measurable subset $A$ of $\R^{n-1}$,
$$\int_A h \, d {\cal L}^{n-1} \le 2^{1\over 2}
{\cal L}^{n-1}(A)^{1 \over 2} \left( \int_{{\bf R}^{n}} f^2 \, d
{\cal L}^{n}\right)^{1 \over 4}
\left( \int_{{\bf R}^{n}} |\nabla f|^2 \, 
d {\cal L}^{n}\right)^{1 \over 4}.$$
\smallskip
{\bf Proof.}  According to Schwarz's inequality 
we have
$$\int_A h \, d {\cal L}^{n-1} \le {\cal L}^{n-1}(A)^{1 \over
2}\left( \int_A h^2 \, d {\cal L}^{n-1}\right)^{1 \over 2},$$
and we can use Schwarz's inequality together with the estimate
from Step 1 to estimate
$$\eqalign{\int_A h^2 \, d {\cal L}^{n-1}
&
\le \int_{x \in {\bf R}^{n-1}} h^2(x) \, d {\cal L}^{n-1}x
\cr
& \le
\left(\int_{x \in{\bf R}^{n-1}}
h^2(x)\, r(x) \, d {\cal L}^{n-1}x\right)^{1 \over 2}
\left(\int_{x \in {\bf R}^{n-1}} {h^2(x)\over r(x)}
\, d {\cal L}^{n-1}x \right)^{1 \over 2}\cr
\cr
& \le
\left(2 \int_{x \in{\bf R}^{n-1}}
\int_{s \in {\bf R}}
f^2(x, s) 
\,d {\cal L}^1 s \,  d {\cal L}^{n-1}x
\right)^{1 \over 2}
\cr
& \phantom{XXXX}
\left(2 \int_{x \in {\bf R}^{n-1}}
\int_{s \in {\bf R}}
\left({\partial f \over \partial r}(x, \, s)\right)^2
\,d {\cal L}^1 s 
\, d {\cal L}^{n-1}x\right)^{1 \over 2}\cr
}$$
from which two inequalities together with Fubini's theorem
our assertion follows.
\bigskip
{\bf Step 3.} Suppose $f \colon \R^n \to \R^+$ is a Lipschitz
function having compact support and $K \subset \R^{n-1} \times
[R, \, S] \subset \R^n$ is a set of finite perimeter.  Then,
for each positive integer $M$,
the oriented surface integral
$$ \Bigg|\int_{p \in \partial K} f(p)\, {\bf n}_K(p) \bullet {\bf
e}_n\, d \HH p\,\Bigg|$$
does not exceed the sum
$$2^{3\over 2} M \, 
{\cal L}^{n-1} \big(\pi \partial K\big)^{1 \over 2}
\left(
\int_{{\bf R}^n} f^2 \, d \LL
\right)^{1 \over 4}
\left(
\int_{{\bf R}^n} |\nabla f |^2 \, d \LL
\right)^{1\over 4}
\phantom{XXXXXXXXXXX}
$$
$$\phantom{XXXXXXXXXXXXXXXXXXXXX}
+ \left(
{(S-R)\HH(\partial K) \over 2M}
\right)^{1 \over 2}
\left(
\int_{{\bf R}^n} |\nabla f|^2\,d \LL
\right)^{1 \over 2}
 $$
\smallskip
{\bf Proof.}
We let $\pi \colon \R^{n-1} \times \R \to \R^{n-1}$
denote projection on the first factor.
We infer from the
theory of slicing set forth in [FH 4.3.8] that, for
${\cal L}^{n-1}$ almost every $x$ in $\R^{n-1}$, there will be
a nonnegative integer $N(x)$ together with numbers
$$R \le r_1(x) < s_1(x) < r_2(x) < s_2(x) < \ldots < r_{N(x)}
<s_{N(x)} \le S$$
such that the slice
$\big< \partial \l K \r , \, \pi, \, x\big>$
can be written
$$\big< \partial \l K \r , \, \pi, \, x\big> =
\sum_{i=1}^{N(x)} \l (x, \, s_i(x))\r - \l (x, \, r_i(x))\r$$
so that
$$\int_{p \in \partial K} f(p)\, {\bf n}_K(p) \bullet {\bf
e}_n\, d \HH p
= \int_{x \in {\bf R}^{n-1}} 
\sum_{i=1}^{N(x)}  f\big(x, \, s_i(x)\big) - f\big(x,\, r_i(x)\big)
\, d {\cal L}x.$$
For each positive integer $M$ we set
$$A_M = \R^{n-1} \cap \{x \colon 1 \le N(x) \le M\}\quad\hbox{and}\quad
B_M = \R^{n-1} \cap \{x \colon N(x) > M\}$$
and we estimate
$$\eqalign{
\Bigg|
\int_{p \in \partial K} f(p)\, {\bf n}_K(p)& \bullet {\bf e}_n\, d \HH p
\,\Bigg|
\cr
&
\le
2M \int_{x \in A_M} \sup \{ \big|f(x, \, r)\big|
\colon R \le r \le S\}\, d {\cal L}^{n-1}x
\cr &  
\phantom{XXXXX}+
\int_{x \in B_M} \int_{t \in \left[R, \, S\right]}
\left|{\partial f \over \partial r}(x, \, t)\right| \, d {\cal
L}^1 \, d {\cal L}^{n-1}x
\cr
&
\le 2^{3 \over 2} M \, 
{\cal L}^{n-1} (\pi \partial K)^{1 \over 2}
\left(
\int_{{\bf R}^n} f^2 \, d \LL
\right)^{1 \over 4}
\left(
\int_{{\bf R}^n} |\nabla f |^2 \, d \LL
\right)^{1\over 4}
\cr 
&
\phantom{XXXXX}+
\LL\big(B_M\times [R, \, S]\big)^{1 \over 2}
\left(
\int_{{\bf R}^n} |\nabla f |^2 \, d \LL
\right)^{1\over 2}
\cr 
&\le
2^{3\over 2}M\,
{\cal L}^{n-1} (\pi \partial K)^{1 \over 2}
\left(
\int_{{\bf R}^n} f^2 \, d \LL
\right)^{1 \over 4}
\left(
\int_{{\bf R}^n} |\nabla f |^2 \, d \LL
\right)^{1\over 4}
\cr 
&
\phantom{XXXXX}
+
\left( {(S-R)\HH\big(\partial K\big)\over 2M}\right)^{1
\over 2}
\left(
\int_{{\bf R}^n} |\nabla f |^2 \, d \LL
\right)^{1\over 2};
\cr}$$
In making the estimates above we have used Step~2 applied to
the function $|f|$, the Fundamental Theorem of Calculus,
the Hausdorff area formula [FH 3.2.20], Schwarz's inequality,
and Fubini's theorem.
\bigskip
{\bf Step 4.}
Let $K$ and $f$ be as in the hypotheses of the present theorem.
Since the support of $f$ lies in the open ball
$\U^n\big(p, \, \rho_\ast/2\big)$ 
it will lie in  some closed ball
$\B^n\big(p, \, \rho_\ast/2 - 2 \epsilon\big)$ 
for some $\epsilon$.  The set
$L = K \cap \U^n\big(p, \, \rho_\ast/2 -  \epsilon\big)$ 
belongs to $\KK$ with
$$ {\cal L}^{n-1}\big(\pi \, \partial L\big) <
\alpha(n)\left({\rho_\ast\over 2}\right)^{n-1}\hbox{ and }
\HH(\partial L)<
\HH (\partial K)+n\alpha(n)\left({\rho_\ast\over 2}\right)^{n-1}.$$
Identification of $\O$ with the quotient of $\R^n$ by
$\rho_\ast$ times the integer lattice enables us to reduce
our problem to a problem in $\R^n$ as in Step~3 in which $K$
there (a single inverse image of $L$) lies in the cube
$\big[0, \, \rho_\ast\big]^n$ and
$f$ there has its support in the same cube.  The assertion of
our theorem now follows from the estimate in Step~3. \E
\bigskip
{\bf 8.3 Theorem (Bounds on the change in surface integrals of
$H(T)$ under heat flow).}
{\sl
There is a constant $C$ with the following property.

\noindent
Suppose
\smallskip\noindent
(a) $\l K_0\r \in \KK$ and $Q_0 \in \QQ$ give an $\EEE$
minimizer for some heat distribution $P$ and some time
interval $\Delta t \le 1$;
\smallskip\noindent
(b) $0 < t_1 < \Delta t$ and $Q(x, \, t)$ and $T(x, \, t)$ are
functions defined on $\O \times [ 0, \, t_1]$ which are a
solution to the heat equation with fixed crystal $\l K_0\r$
and initial heat distribution $Q_0$ and final heat
distribution~$Q_1$ as in 2.5, 5.1, 5.2;
\smallskip\noindent
(e) $T_0 = c_{K_0}^{-1}Q_0$ is the temperature field associated
with $\l K_0\r$ and $Q_0$ satisfying the {\it a priori}
estimates of~4.2,
and  $T_1 = c_{K_0}^{-1}Q_1$ is the temperature field associated
with $\l K_0\r$ and $Q_1$;
\smallskip\noindent
(d) $g \colon \O \to \R$ is a smooth function having support
in an open ball $\,\U^n\big(p, \, \rho_\ast/2\big)$.
\smallskip\noindent
Then, for each $i = 1, \, 2, \, \ldots,\, n$,
$$\Bigg|\int_{x \in \partial K_0} H(T_0(x))\, g(x)\, {\bf
n}_K(x)\bullet {\bf e}_i \, d \LL x
-\int_{x \in \partial K_0} H(T_1(x))\, g(x)\, {\bf n}_K(x)
\bullet {\bf e}_i \, d \LL x \Bigg|$$
$$\phantom{XXXXXXXXXXXXXXXXXXXXXXXX}
\le C \Delta t^{1/12 - 2 \alpha}\sup\bigg\{\sup_\O |g|, \,
\sup_\O|Dg|\bigg\}.$$
}
\bigskip
PROOF.
It clearly suffices to treat only the case in which $i = n$
which assumption we make.
The basic calculations in our proof are in terms of our
time interval $\Delta t$ raised to various powers.
In order to simplify the exposition we will keep accurate
accounting of the exponents of $\Delta t$ but will denote by $C$ a
general (large) constant which will possibly have different
values, even in the same expression.
We will prove our theorem by invoking Theorem~8.2 with
function
$$f(x) = \Big[H\big(T_0(x)\big)- H\big(T_1(x)\big)\Big]\,
g(x)$$
and with constant
$$M = \Delta t^{-{1 \over 6}}.$$
From Theorem~4.3 we know that, for each $x$,
$$C^{-1} \Delta t^{n\alpha \over 3n+2} \le T_0(x) \le C +
C \Delta t^{-\alpha} \le C \Delta t^{-\alpha}.\leqno{(i)}$$
The maximum principle for solutions to the heat equation 
together with the corresponding minimum principle
imply that
$$C^{-1} \Delta t^{n\alpha \over 3n+2} \le T_1(x) \le	
C \Delta t^{-\alpha}.\leqno{(ii)}$$
We infer from Theorem~4.3 that $T_0$ satisfies a
Lipschitz condition with constant $C \Delta t^{-\alpha}$ so
that
\smallskip\noindent
(iii) $\dir(T_0) \le C\Delta t^{-2\alpha}$.
\smallskip\noindent
We use conclusion (4) of Theorem~5.2 to infer that
\smallskip\noindent
(iv) $\dir(T_1) \le C\Delta t^{-2\alpha}$.
\smallskip\noindent
We use conclusion (5) of Theorem~5.2 to infer
$$\int_\O \big|T_0 - T_1\big|^2 \, d \LL \le C \Delta t^{1 - 2
\alpha}.\leqno{(v)}$$
We use estimate A.1.14 of Appendix~A.1 together with
(i) above to infer, for each $x$,
$$\eqalign{\Big|H\big(T_0(x)\big)& - H\big(T_1(x)\big)\Big|
\cr &
\le C \big|T_0(x) - T_1(x)\big|\,
\left({1 \over T_0(x) T_1(x)^2}+ {1 \over T_0(x)^2 T_1(x)} + T_0(x) +
T_1(x)\right)
\cr &
\le C \big|T_0(x) - T_1(x)\big|
\left({1 \over \Delta t^{3 n\alpha \over 3n+2}} + {1 \over \Delta
t^\alpha} \right)
\cr &
\le C  \Delta t^{-\alpha} \big|T_0(x) - T_1(x)\big|.}\leqno{(vi)}$$
Similarly we use estimate A.1.7 of Appendix~A.1 together with
(i) above to infer, for each~$x$,
$$\eqalign{\Big|H'\big(T_0(x)\big)\nabla T_0(x)
- H'\big(T_1(x)\big)&\nabla T_1(x) \Big|
\cr &
\le C \bigg(\big|\nabla T_0(x) \big| + \big|\nabla T_1(x)\big|\bigg)
\left({1 \over \Delta t^{3 n\alpha \over 3n+2}} + {1 \over \Delta
t^\alpha} \right)
\cr &
\le C \bigg(\big|\nabla T_0(x) \big| + \big|\nabla T_1(x)\big|\bigg)
\Delta t^{- \alpha}. \cr}\leqno{(vii)}$$
We combine (v) and (vi) above to estimate
$$\int_\O f^2 \, d \LL \le \sup_\O g^2\, \Delta t^{1 - 4
\alpha}\leqno{(viii)}$$
and combine (i), (iii), (iv),  (v),
and (vii) above together with the chain rule to estimate
$$\int_\O|\nabla f|^2 \, d \LL
\le C \sup_\O g^2 \, \Delta t^{-4\alpha}
+ C \sup_\O |\nabla g|^2 \, \Delta t^{1 - 4\alpha}
\le C \sup\big\{ \sup_\O g^2, \,\sup_\O|\nabla g|^2 \big\}
\Delta t^{- 4
\alpha}\leqno{(ix)}$$
We now use 8.2 to estimate
$$\Big|\int_\O f \, {\bf n}_{K_0} \bullet {\bf e}_n\, d \LL \Big|
\le C \Delta t^{- {1 \over 6}}\, \Delta t^{{1 - 4 \alpha} \over
4} \, \Delta t^{- {4\alpha \over 4}}
+  C \Delta t^{1 \over 12} \, \Delta t^{- {4 \alpha\over 2}}
= C \Delta t^{{1 \over 12 } -2 \alpha}.$$
The theorem follows.  \E
\bigskip
{\bf 8.4 Terminology.} 
\smallskip\noindent
(1) For each $\epsilon > 0$ we set
$$H_\epsilon(T) 
= \cases{H(\epsilon) &if $0 < T < \epsilon$\cr
\phantom{X} & \cr
         H(T)        &if $\epsilon \le T \le {1 \over \epsilon}$\cr
\phantom{X} & \cr
	 H\left({1 \over \epsilon}\right)
	 &if ${1 \over \epsilon} < T < \infty$.\cr}$$
\smallskip \noindent
(2) We say that a function $T\colon \O \to \R$ is {\bf
strictly defined} if  and only if 
$$T(p) = \lim_{r \downarrow 0} \alpha(n)^{-1} r^{-n}
\int_{{\bf B}^n(p, \, r)} T \, d\LL$$
at each point $p$ at which the limit above exists.
\bigskip
{\bf 8.5 Theorem (An approximate Stokes theorem for the \wmc~
integral).}
{\sl
Suppose that $t$ is one of the almost all positive times for
which crystals, heat distributions, and temperature fields
converge as in conclusion (1) of Theorem 7.7. 
Suppose also that $T_\ast(t)$ is strictly defined.  Then
$$ \lim_{\epsilon \downarrow 0}
\int_{x \in \partial K_\ast(t)} H_\epsilon \big(T_\ast(x, \, t)\big)
\, {\bf n}_{K_\ast(t)}(x)\bullet g_\ast(x)\, d \HH x\phantom{XXXX}$$
$$\phantom{XXXXXXXX}
= \int_{x \in K_\ast(t)} \div\left( H\big(T_\ast(x, \, t)\big)
\, g_\ast(x)\right)\, d \HH x$$
for each smooth vector field $g_\ast \colon \O \to \R^n$.
}
\bigskip
PROOF.  It clearly suffices to consider only the case $g_\ast
= g\, {\bf e}_n$ for some smooth function $g\colon \O \to \R$.
In view of B.7 we infer from our hypotheses
that $\dir(T_\ast(t)) < \infty$.
We infer  from [MC Lemma 3.1.1] that the functions
$$x_n \mapsto T_\ast\big(x_1, \, \ldots, \,  x_{n-1}, \, x_n, \,
t\big)$$
are absolutely continuous for
${\cal L}^{n-1}$ almost all $\big(x_1, \, \ldots, \, x_{n-1}\big)$
(with the obvious meaning in $\O$).
One uses a representation of $\partial \l K_\ast(t)\r$ in
terms of slices as in the proof of Step 3 in the proof 
of Theorem~8.2 (note Step~4 there) together with the
fundamental theorem of calculus to infer
$$ \int_{x \in \partial K_\ast(t)} H_\epsilon \big(T_\ast(x, \, t)\big)
\, g(x) \, {\bf e}_n \bullet
\, {\bf n}_{K_\ast(t)}(x)\, d \HH x$$
$$\phantom{XXXXXXXX}
= \int_{x \in K_\ast(t)} {\partial\over \partial x_n}
\left( H_\epsilon\big(T_\ast(x, \, t)\big)
\, g(x)\right)\, d \HH x.$$
We check that the the integral
$$\int_{x \in K_\ast(t)} \div\left( H\big(T_\ast(x, \, t)\big)
\, g_\ast(x)\right)\, d \HH x$$
exists and is finite and use Lebesgue's bounded convergence
theorem to conclude that
$$\lim_{\epsilon \downarrow 0}
\int_{x \in K_\ast(t)} \div\left( H_\epsilon\big(T_\ast(x, \, t)\big)
\, g_\ast(x)\right)\, d \HH x
= \int_{x \in K_\ast(t)} \div\left( H\big(T_\ast(x, \, t)\big)
\, g_\ast(x)\right)\, d \HH x.  $$
The theorem follows.  \E
\bigskip
{\bf 8.6 Theorem (Minimizing properties of $\l K_\ast(t)\r$'s).}
{\sl
Suppose that $t$ is one of the almost all positive times for
which the crystals, heat distributions, and temperature fields
converge as in conclusion~(1) of Theorem~7.7. Then
$$\EEE\big(\l K_\ast(t)\r, \, Q_\ast(t)\big) = \inf\bigg\{
\EEE\big(\l K \r, \, Q_\ast(t)\big)\colon \l K \r \in \KK\bigg\};
\leqno{(1)}$$
$$\Phi\big(\partial \l K_\ast(t) \r \big) = \lim_{i \to
\infty}\Phi\big(\partial \l K_{J(i)}(t) \r \big);\leqno{(2)}$$
If we assume additionally that $\Phi$ is smooth and elliptic the we
further conclude
$$\M\big(\partial \l K_\ast(t) \r \big) = \lim_{i \to
\infty}\M\big(\partial \l K_{J(i)}(t) \r \big);\leqno{(3)}$$
$${d \over ds}\Phi\Big( G_{s\sharp} \partial \l K_\ast(t)\r\Big)
\Bigg|_{s = 0}
=\lim_{i \to \infty}
{d \over ds}\Phi\Big( G_{s\sharp} \partial \l K_{J(i)}(t)\r\Big)
\Bigg|_{s = 0}\leqno{(4)}.$$
for each (fixed) smooth vector field $g \colon \O \to \R^{n}$
and associated deformation of $\O$ given by setting
$G_t(x) = x + s\, g(x)$ for small times $s$.
}
\bigskip
PROOF. 
For each large $i$ we set
$\Delta t_i = 2^{-J(i)}$ (so that $\Delta t_i$ is the time
step used in constructing the $J(i)$-th approximating
evolution) and denote by $k(i)$ the unique 
positive integer for which
$$k(i) \Delta t_i \le t < (k(i)+1) \Delta t_i$$
$k(i)\Delta t_i$ is thus the beginning of the interval in which
the time $t$ appears in constructing the $J(i)$-th approximating
evolution.
The idea in proving (1) and (2) is to assume the
negation and show that this will violate the property
$$\EEE\big(\l K_{J(i)}(k(i)\Delta t_i)\r,
\, Q_{J(i)}(k(i) \Delta t_i)\big) = \inf\bigg\{
\EEE\big(\l K\r, \, Q_{J(i)}(k(i)\Delta t_i)\big)
\colon \l K \r \in \KK\bigg\}$$
which holds by construction (see~2.11.4).
There is a lot of terminology involved here, and it seems useful
to make abbreviations in the proof.  

For large $i$ we further set
$$K_i = K_{J(i)}(t) = K_{J(i)}(k(i) \Delta t_i) \quad
K_\ast=K_\ast(t)$$
and denote by $\l K \r$ a general (but fixed) crystal.
Additionally we set
$$c_i = c_{K_i}, \quad c_\ast = c_{K_\ast}, \quad c = c_{K}$$
$$P_i = Q_{J(i)}(k(i)\Delta t_i), \quad
Q_i = Q_{J(i)}(t), \quad Q_\ast = Q_{\ast}(t)$$
$$S_i = T_{J(i)}(k(i)\Delta t_i), \quad
T_i = T_{J(i)}(t), \quad T_\ast = T_{\ast}(t)$$
(our use of ``$P$'' here differs from the usage in 2.11.3).
We will denote by $C$ a general (large) constant (which is
independent of $i$ and $K$) which will possibly have different
values in the same  expression.  The remainder of the  proof
occurs in six steps.
\bigskip
{\bf Step 1.}  According the conclusion~(1) of Theorem~4.3 we
know
$$C^{-1} \Delta t_i^{n\alpha \over 3n+2} \le S_i \le C \Delta
t_i^{-\alpha}.\leqno{(i)}$$
The maximum principle for solutions to the heat equation
together with the
corresponding minimal principle imply that also
$$C^{-1} \Delta t_i^{n\alpha \over 3n+2} \le T_i \le C \Delta
t_i^{-\alpha}.\leqno{(ii)}$$
We infer from conclusion~(2) of Theorem~4.3 that each $S_i$ satisfies a
Lipschitz condition with constant $C \Delta t_i^{-\alpha}$ so
that
\smallskip\noindent
(iii) $\dir(S_i) \le C\Delta t_i^{-2\alpha}$.
\smallskip\noindent
We use conclusion (4) of Theorem~5.2 to infer that
\smallskip\noindent
(iv) $\dir(T_i) \le C\Delta t_i^{-2\alpha}$.
\smallskip\noindent
We use conclusion (5) of Theorem~5.2 to infer
$$\int_\O \big|S_i - T_i\big|^2 \, d \LL \le C \Delta t_i^{1 - 2
\alpha}.\leqno{(v)}$$
\bigskip
{\bf Step 2.} For each $i$,
$$\Bigg|\int_\O c F\big(c^{-1} P_i\big)\,d \LL
- \int_\O c F\big(c^{-1} Q_i\big)\,d \LL \Bigg| \le C \,
\Delta t_i^{{1 \over 2} - 2 \alpha}.$$
\medskip
{\bf Proof.}  We use the extreme inequality of A.1.13 to infer
that the difference of the integrals on the left above does
not exceed
$$C \int_\O \bigg| c^{-1}P_i - c^{-1}Q_i \bigg| \left(
{c^3 \over P_i^2 Q_i}
+{c^3 \over P_i Q_i^2}
+ {P_i \over c}
+ {Q_i \over c} \right)\, d \LL$$
which in turn does not exceed
$$C \int_\O \bigg| S_i - T_i \bigg| \left(
{1 \over S_i^2 T_i}
+{1 \over S_i T_i^2}
+ {S_i}
+ {T_i} \right)\, d \LL.$$
We use estimates (i) and (ii) in Step~1 above to dominate the
right hand factor  in the integrand above.  We then use Schwarz's
inequality together with estimate (v) in Step~1 above to
dominate the contribution of the  left hand side.  The
assertion of Step~2 follows.
\bigskip
{\bf Step 3.}  We assert
$$\lim_{i \to \infty} \int_\O c_i F\big(T_i\big)\, d \LL =
 \int_\O c_\ast F\big(T_\ast\big)\, d \LL.$$
\medskip
{\bf Proof.}  We write
$$\int_\O c_i F\big(T_i\big)
 - c_\ast F\big(T_\ast\big)\, d \LL
=\int_\O (c_i - c_\ast) F\big(T_\ast\big)\, d \LL
+ \int_\O c_i\bigg( F\big(T_i\big)
- F\big(T_\ast\big)\bigg)\, d \LL.$$
The left hand integral on right side of the equality
immediately above converges to  zero as
$i \to \infty$ as a consequence the convergence of the
crystals $\l K_i \r$ to $\l K_\ast \r$ and the absolute
continuity of integrals of summable functions.
The right hand integral there above converges to
zero as a consequence of the left hand inequality of A.1.13
together with the strong convergence of
temperature fields $T_i$ to $T_\ast$ and of reciprocal
temperature fields $1/T_i$ to $1/T_\ast$. 
\bigskip
{\bf Step 4.} We assert
$$\lim_{i \to \infty} \int_\O c F\big(c^{-1}Q_i\big)\, d \LL =
 \int_\O c F\big(c^{-1}Q_\ast\big)\, d \LL.$$
\medskip
{\bf Proof.}  We use the left hand inequality of A.1.13 to
estimate
$$\eqalign{\bigg| c F\big(c^{-1}Q_i\big)
    - c  F\big(c^{-1}Q_\ast\big)\bigg|
&\le C \Bigg(\left|{1 \over Q_i^2} - { 1 \over Q_\ast^2}\right|
+ \left|Q_i - Q_\ast\right|\Bigg)
\cr &
\le C \Bigg(\left|{c_i^2 \over Q_i^2} 
    - { c_\ast^2 \over Q_\ast^2}\right|
+ \left|\left({c_\ast^2 \over c_i^2 } - 1\right){c_i^2\over
Q_i}\right|
\cr &
\phantom{XXXX}
+ \left|{Q_i\over c_i} - {Q_\ast\over c_\ast} \right|
+ \left|\left({ c_i \over c_\ast} - 1\right)
{Q_i\over c_i}\right|\Bigg).\cr}$$
The integrals of the first and third summands immediately above
converge to zero as $i \to \infty$ as a consequence of
the strong convergence of
temperature fields $T_i$ to $T_\ast$ and of reciprocal
temperature fields $1/T_i$ to $1/T_\ast$. 
\bigskip
{\bf Step 5.}  The negation of the first  assertion of our
theorem would imply the existence of a crystal $\l K \r$ such
that
$$\Phi\big(\partial \l K \r\big) + \int_\O c F\big(c^{-1} Q_\ast\big)
<\Phi\big(\partial \l K_\ast \r\big) + \int_\O c_\ast F\big(c_\ast^{-1}
Q_\ast\big).$$
The reason this does not happen is that, for large~$i$'s,
$\l K \r$ would then
also be a better choice than $\l K_i \r$ in the minimization
at the $i$-th step which occurred at time $k(i)\Delta t_i$, i.e.
we would have
$$\Phi\big(\partial \l K \r\big) + \int_\O c F\big(c^{-1} P_i\big)
<\Phi\big(\partial \l K_i \r\big) + \int_\O c F\big(c_i^{-1}
P_i\big).$$
This last inequality holds for large~$i$ as a consequence of
the first inequality by virtue of the bulk integral
estimate in Step~2 together with
the convergence of bulk integrals asserted in Steps~3 and~4
and the lower semicontinuity of interface integrals, i.e. the
fact that $\Phi\left(\l K_\ast \r \right) \le \lim \inf_i 
\Phi\left(\l K_i \r \right)$.
The negation of the second assertion of our theorem would
imply
$\Phi\left(\l K_\ast \r \right) < \lim \sup_i 
\Phi\left(\l K_i \r \right)$.
An argument similar to that above would show that, 
for arbitrarily large~$i$'s, $\l K_\ast \r$ would then
also be a better choice than $\l K_i \r$ for the minimization
at the $i$-th step which occurred at time $k(i)\Delta t_i$, i.e.
$$\Phi\big(\partial \l K_\ast \r\big)
+ \int_\O c_\ast F\big(c_\ast^{-1} P_i\big)
<\Phi\big(\partial \l K_i \r\big) + \int_\O c_i F\big(c_i^{-1}
P_i\big).$$
\bigskip
{\bf Step 6.}  We now consider assertions (3) and (4) of our theorem
above.  Under the assumption that $\Phi$ is smooth and
elliptic we infer from Conclusion~(2) of the present theorem
together with Theorem~C.3 in Appendix~C that there is
weak varifold convergence
$${\bf v}\bigg( \partial K_{J(i)}(t)\, , \,1\, , \, {\bf
n}_{K_{J(i)}(t)}\bigg) \h
{\bf v}\bigg( \partial K_\ast(t)\, , \,1\, , \, {\bf
n}_{K_\ast(t)}\bigg)$$
as $i \to \infty$ in the sense of the terminology set forth in
C.1(6).
This varifold convergence immediately implies Conclusion~(3)
of the present theorem.  The varifold convergence together
with Theorem~C.2 imply Conclusion~(4) of the present theorem.
\E
\bigskip
{\bf 8.7 Theorem (The Gibbs-Thomson condition holds in the
limit in  the sense of first variation distributions when
$\Phi$ is smooth and elliptic).}
{\sl
Suppose that $\Phi$ is smooth and elliptic and
that $t$ is one of the almost all positive times for
which the crystals, heat distributions, and temperature fields
converge as in conclusion~(1) of Theorem~7.7. Then
$$\eqalign{{d \over ds} \Phi\bigg(G_{s\sharp}\partial \l K_\ast(t) \r
\bigg)\Bigg|_{s=0}
&
= \lim_{\epsilon \downarrow 0}
\int_{x \in \partial K_\ast(t)} H_\epsilon \big(T_\ast(x, \, t)\big)
\, {\bf n}_{K_\ast(t)}(x)\bullet g_\ast(x)\, d \HH x
\cr &
= \int_{x \in K_\ast(t)} \div\left( H\big(T_\ast(x, \, t)\big)
\, g_\ast(x)\right)\, d \LL x\cr }$$
for each smooth initial velocity vector field $g_\ast \colon \O \to
\R^n$ and associated deformation $G_s(x) = x + s\, g_\ast(x)$ for
small $s$.
}
\bigskip
PROOF.  Since both sides of the asserted equation are linear
in $g_\ast$ [AW 4.9], it is clearly sufficient to prove the
assertion in case $g_\ast = g\, {\bf e}_n$ for some smooth
function $g$ having small support in $\O$.  We fix such a
$g$.  We also fix a subsequence of the $J(i)$'s, which we will simply
index by the subscript $j$, for which $\nabla T_j(t)\h \nabla
T(t)$ and $\nabla U_j(t) \h \nabla U(t)$ as $j \to \infty$
with $\sup_j \dir\big(T_j(t)\big) < \infty$ and
$\sup_j \dir\big(U_j(t)\big) < \infty$.
This is possible by 7.7(1) and B.7.
We set $\Delta t_j = 2^{-J(i)}$ as the time step used in
constructing the $J(i)$-th approximating evolution associated
with the present subscript~$j$.  We further denote by $k(j)$ 
the unique positive integer for which
$$k(j)\, \Delta t_j \le t < \big(k(j)+ 1\big)\, \Delta t_j$$
so that $k(j) \Delta t_j$ is the beginning of the time
interval in which $t$ appears.
According to Theorem~4.4,
$${d \over ds} \Phi\bigg(G_{s\sharp}
\partial \l K_j(k(j)\Delta t_j) \r \bigg)\Bigg|_{s=0}$$
equals
$$\int_{x \in \partial K_j(k(j)\Delta t_j)}
H\big(T_j(x, \, k(j)\Delta t_j)\big)
\, g(x) \,{\bf n}_{K_j(k(j) \Delta t_j)}(x)
\bullet {\bf e}_n\, d \HH x.$$
According to conclusion (4) of Theorem~8.4,
$${d \over ds} \Phi\bigg(G_{s\sharp}
\partial \l K_j(k(j)\Delta t_j) \r \bigg)\Bigg|_{s=0}
= {d \over ds} \Phi\bigg(G_{s\sharp}
\partial \l K_j(t) \r \bigg)\Bigg|_{s=0}
 \to {d \over ds} \Phi\bigg(G_{s\sharp}
\partial \l K_\ast(t) \r \bigg)\Bigg|_{s=0}$$
as $j \to \infty$.
According to Theorem~8.3 the difference between
$$\int_{x \in \partial K_j(k(j)\Delta t_j)}
H\big(T_j(x, \, k(j)\Delta t_j)\big)
\, g(x) \,{\bf n}_{K_j(k(j) \Delta t_j)}(x)
\bullet {\bf e}_n\, d \HH x.$$
and
$$\int_{x \in \partial K_j(t)}
H\big(T_j(x, \, t)\big)
\, g(x) \,{\bf n}_{K_j(t)}(x)\bullet {\bf e}_n\, d \HH x$$
converges to zero as $j \to \infty$ since $\Delta t_j \to 0$
and $\alpha < 1/24$, in accordance with 2.10.
According to
the general Gauss-Green Theorem [FH 4.5.6(5)]
$$\int_{x \in \partial K_j(t)}
H\big(T_j(x, \, t)\big)
\, g(x) \,{\bf n}_{K_j(t)}(x)\bullet {\bf e}_n\, d \HH x$$
equals
$$\int_{x \in  K_j(t)}
{\partial \over \partial x_n}\Bigg(H\big(T_j(x, \, t)\big)
\, g(x) \Bigg)\, d \HH x$$
and we know from Theorem~8.5 that
$$\lim_{\epsilon\downarrow 0}\int_{x \in \partial K\ast(t)}
H_\epsilon\big(T_\ast(x, \, t)\big)
\, g(x) \,{\bf n}_{K_\ast(t)}(x)\bullet {\bf e}_n\, d \HH x$$
equals
$$\int_{x \in  K_\ast(t)}
{\partial \over \partial x_n}\Bigg(H\big(T_\ast(x, \, t)\big)
\, g(x) \Bigg)\, d \HH x$$
According to Theorem~8.1
$$\int_{x \in  K_j(t)}
{\partial \over \partial x_n}\Bigg(H\big(T_j(x, \, t)\big)
\, g(x) \Bigg)\, d \HH x$$
converges to
$$\int_{x \in  K_\ast(t)}
{\partial \over \partial x_n}\Bigg(H\big(T_\ast(x, \, t)\big)
\, g(x) \Bigg)\, d \HH x$$
as $j \to \infty$.
The theorem follows. \E
\bigskip
{\bf 8.8 Theorem (Minimality and regularity of the limit crystals
in low dimension when $\Phi$ is smooth and elliptic).}
{\sl
Suppose that our
surface energy integrand $\Phi$ is smooth and elliptic and
that $t$ is one of the almost all positive times for
which the crystals, heat distributions, temperature fields,
and reciprocal temperature  fields
converge to limits $\l K_\ast(t)\r$, $Q_\ast(t)$, $T_\ast(t)$,
$U_\ast(t)$
as in conclusion~(1) of Theorem~7.7 or as in Theorem~8.6. Then
\smallskip\noindent
(1) If the ambient dimension $n$ equals 2 or 3 then there is a
function $\omega \colon (0, \, 1) \to \R^+$ with $\lim_{s
\downarrow 0} \omega(s) = 0$ together with a positive
number~$\delta$ such that $\partial \l K(t)\r$ is $\big( \Phi,
\, \omega, \, \delta \big)$-minimal in the sense of Bombieri
[BE Definition~1, page 101].
This minimality means means that whenever $W$ is a compact
subset of $\O$ with small  diameter and $Q$ is an
$n$~dimensional integral current in~$\O$ having support in~$W$
with $r = \hbox{diam}(W) \le \delta$ then
$$\Phi\big(\partial \l K_\ast(t) \r \rightangle W \big) \le
\Phi\big(\partial \l K_\ast(t) \r \rightangle W + \partial Q \big)
+ \omega (r)\, \M\big(\partial \l K_\ast(t)\r \rightangle W +  Q\big).$$
\smallskip \noindent
(2) In case $n = 2$ then
$$\int_{(0, \, 1)} {\omega(s)^{1 \over 2}\over s}\, d {\cal L}^1 <
\infty$$
from which it follows that $\partial K_\ast(t)$
is a one dimensional differentiable submanifold of $\O$
without boundary
and
$${d \over ds} \Phi\bigg(G_{s\sharp}\partial \l K_\ast(t) \r
\bigg)\Bigg|_{s=0}
= \int_{x \in \partial K_\ast(t)} H \big(T_\ast(x, \, t)\big)
\, {\bf n}_{K_\ast(t)}(x)\bullet g_\ast(x)\, d \HH x
$$
for each smooth initial velocity vector field $g_\ast \colon \O \to
\R^n$ and associated deformation $G_s(x) = x + s\, g_\ast(x)$ for
small $s$.
\smallskip \noindent
(3) In case $n=3$ then $\partial K_\ast(t)$ is the
homeomorphic image in $\O$ of a compact two dimensional manifold
without boundary. 
}
\bigskip
PROOF.  
Our proof is based on the regularity theorem of
almost minimal currents [BE Main Theorem, page~102] together with 
Reifenberg's topological disk theorem [MC Theorem~10.5.1,
page~439] with use of the regularity estimate of [ASS].  We abbreviate
$K = K_\ast(t)$, $Q = Q_\ast(t)$, $T = T_\ast(t)$, $U =
U_\ast(t)$.
In the following we denote by $C$ a general constant whose
value may be different at different times (even in the same
equation) which depends only on $n$, $\rho$ and upper bounds for the
integral
$$\int_\O U^2 + |\nabla U|^2 + T^2 + |\nabla T|^2 \, d \LL.$$
\medskip
{\bf Step 1. Some preliminary estimates.}
As in [ATW 3.1.4] we can write
$$\l K \r + Q = {\bf t}\big(A, \, \theta, \, \sigma\big)
= \sum_{i=1}^\infty \l P_i \r - \l N_i \r$$
where
$$P_i = \big\{ x \colon \sigma(x) = {\bf e}_1 \wedge \ldots
\wedge {\bf e}_n \hbox{ and } \theta(i) \ge i\big\}, \quad
N_i = \big\{ x \colon \sigma(x) = -{\bf e}_1 \wedge \ldots
\wedge {\bf e}_n \hbox{ and } \theta(i) \ge i\big\} $$
so that $P_1 \cap N_1 = \emptyset$ and
$P_1 \supset P_2 \supset P_3 \supset \ldots$ and
$N_1 \supset N_2 \supset N_3 \supset \ldots$
and
$$\Phi\bigg(\partial \big(\l K \r + Q \big)\bigg)
= \sum_i \Phi\big( \partial \l P_i\r\big) = 
\sum_i \Phi\big(- \partial \l N_i\r\big)$$
and
$$\M\bigg(\partial \big(\l K \r + Q \big)\bigg)
= \sum_i \M\big( \partial \l P_i\r\big) + 
\sum_i \M\big(\partial \l N_i\r\big).$$
If we set $L = P_1$ then
$$\Phi\big(\partial \l L \r \big) \le
\Phi\bigg(\partial \big(\l K \r + Q \big)\bigg)$$
and
$$\M\big(\partial \l L \r \big) \le
\M\bigg(\partial \big(\l K \r + Q \big)\bigg).$$
We abbreviate
$ |K \tt L| =\LL\big( K \tt L \big)$
check that
$| K \tt L|  =
\M\big( \l K \r - \l L \r \big)$
and that
$\partial \big( \l K \r - \l L \r \big)
= \partial  \l K \r \rightangle W  
-\partial \l L \r \rightangle W $
so that, by the isoperimetric inequality [ATW 3.1.7],
$$\eqalign{| K \tt L|
&
\le
{1 \over n^{n \over n-1} \alpha(n)^{1 \over n-1}}\,
\M\Big(\partial \big( \l K \r \rightangle W 
- \partial \l L \r \rightangle W \big)\Big)^{n \over n-1}
\cr &
\le
{1 \over n^{n \over n-1} \alpha(n)^{1 \over n-1}}\,
\Big(\M\big(\partial  \l K \r \rightangle W \big)
+ \M\big(\partial \l L \r \rightangle W \big)\Big)^{n \over n-1}.}$$
\medskip
{\bf Step 2. More preliminary estimates.}  We set
$$\sigma = \Bigg(|K \tt L|^{1/n} + |K \tt L|^{3-n\over
3n}\left( \int_{K \tt L} U^6 + T^6 \, d \LL \right)^{1 \over
3}\Bigg)$$
In case $n = 2$ we use Proposition~B.10 in Appendix~B to
infer
$$ \sigma \le C\, | K \tt L |^{1\over 6} \le C r^{1\over 3}.$$
In case $n=3$ we use B.10 again to infer
$$\sigma \le |K \tt L|^{1\over 3} + C \int_{K \tt L}
U^2 + |\nabla U|^2 + T^2 + |\nabla T|^2\, d \LL$$
and then use the absolute continuity of integrals [FH 2.4.11,
page~85] to infer that $\sigma$ converges to~0 as the
magnitude of $|K \tt L|$ goes to~0.
In particular $\sigma$ converges to~0 as $r$ converges to~0.
We now use inequality A.1.8 in Appendix~A together with
H\"older's inequality to help estimate
$$\eqalign{\int_{K \tt L} c_L
&
F\big(Q/c_L\big) \, d \LL
\cr &
\le
C \int_{K \tt L} 1 + U^2 + T^2 \, d \LL
\cr &
\le
C\Bigg( |K \tt L| +\left(\int_{K \tt L}U^{6} +
T^{6}\, d \LL\right)^{1 \over 3}
|K \tt L|^{2 \over 3}\Bigg)
\cr &
\le
C\,|K \tt L|^{n-1\over n} \Bigg(|K \tt L|^{1/n} + |K \tt L|^{3-n\over
3n}\left( \int_{K \tt L} U^6 + T^6 \, d \LL \right)^{1 \over
3}\Bigg)
\cr &
=
C\, |K \tt L|^{n-1\over n} \sigma
\cr &
\le
C \,
\sigma\,\Big(\M\big(\partial  \l K \r \rightangle W \big)
+ \M\big(\partial \l L \r \rightangle W \big)\Big)
\cr &
\le
C \,
\sigma\,\Big(\Phi\big(\partial  \l K \r \rightangle W \big)
+ \Phi\big(\partial \l L \r \rightangle W \big)\Big)
\cr}
$$
Our $\EEE$ minimality implies
$$\Phi\big( \partial \l K \r\big) + \int_\O c_K\big(Q/c_K)\, d
\LL \le \Phi\big( \partial \l L \r\big) + \int_\O c_L\big(Q/c_L)\, d
\LL$$
from which we infer
$$\eqalign{\Phi\big(\partial \l K \r \rightangle W \big)
&
\le
\Phi\big( \partial \l L \r \rightangle W\big)
+ \int_\O c_L F\big( Q/c_L\big) - c_K F\big(Q/c_K\big)\, d \LL
\cr &
\le
\Phi\big( \partial \l L \r \rightangle W\big)
+ \int_{K \tt L} c_L F\big(Q/c_L\big) \, d \LL
\cr &
\le
\Phi\big( \partial \l L \r \rightangle W\big)
+
C \,
\sigma\,\Big(\Phi\big(\partial  \l K \r \rightangle W \big)
+ \Phi\big(\partial \l L \r \rightangle W \big)\Big)
\cr}$$
Hence
$$(1 - C\sigma)\,\Phi\big( \partial \l K \r \rightangle W \big)
\le (1 + C\sigma)\,\Phi\big( \partial \l L \r\rightangle W \big) $$
so that, in case $r$ and hence $|K \tt L|$ is sufficiently
small
$$\eqalign{\Phi\big( \partial \l K \r\rightangle W\big)
&
\le (1 + 3C\sigma)\,\Phi\big( \partial \l L \r\rightangle W\big)
\cr &
\le \Phi\big( \partial \l L \r\rightangle W\big)
+ C\sigma\,\M\big( \partial \l L \r\rightangle W\big)
\cr &
\le \Phi\big( \partial \l K \r \rightangle W + \partial Q\big)
+ C\sigma\,\M\big( \partial \l K \r \rightangle W + \partial Q\big)
.\cr}$$
We now choose $\omega(r)$ so that $\omega(r) \ge C \sigma$
while $\lim_{s\downarrow 0}\omega(s) = 0$ and the integral
inequality of conclusion~(2) above holds in case $n=2$.
This gives the asserted minimality of conclusion~(1). 
Conclusion~(2) is a consequence of our estimate on $\sigma$
above the the Main Theorem in~[BE].
The first variation formula in conclusion~2 is a consequence
of Theorem~8.7 together with the estimates of [SE
definition, page 135; Proposition~1, page~122; further
results~4.2 and~4.3, page 192,~193] which guarantee the
$\HH$ summability of the restriction of $H(T_\ast(t))$ to
$\partial~K_\ast(t)$.
The argument to obtain the tangent cone criterion of
[BE Remark, page~125] there closely parallels the tangent cone argument
given in [ATW Theorems~3.9 and~3.10] based on the 1~dimensional
regularity of [FH 5.3.10, page~617]; details are left to the
reader.  Conclusion~(3) is a consequence of Reifenberg's theorem  noted
above.  The argument to show there that the closure of $\partial K$
satisfies the $(\epsilon, \, R_1)$ condition there closely
parallels the tangent cone argument given in [ATW Theorems~3.9
and~3.10] based on the 2~dimensional regularity of
[ASS Corollary~3.2, page 255]; details are left to the
reader.
\bigskip
{\bf 8.9 Remark.}  The $C^0$ regularity asserted in
Theorem~8.8 for crystal
boundaries when $n = 3$ is weaker than the $C^1$ regularity
asserted in [LS Section~3, page 106] when $\Phi$ is the
2~dimensional area integrand.
\bigskip
\centerline{\larger Appendix A.  The bulk energy density
function F}

\centerline{\larger and the Gibbs-Thomson function H}
\bigskip 
{\bf A.1 Proposition (Facts about $F$ and $H$).}
{\sl
There is a positive constant $C_\ast$ with the following property.
Suppose $T, \, T_1, \, T_2$ are positive numbers with $T_1 <
T_2$ and $U= 1/T, \, U_1= 1/T_1, \, U_2= 1/T_2$.
}
\smallskip\noindent
$$F(T) = \cases{ U^2 & if $0 < T \le a_\ast$ \cr
                \phantom{x} & \cr
                 T^2 & if $b_\ast \le T < \infty$. \cr}\leqno{\bf A.1.1}$$

$$F'(T) = \cases{ -2 U^3 & if $0 < T \le a_\ast$ \cr
                \phantom{x} & \cr
                 2T & if $b_\ast \le T < \infty$. \cr}\leqno{\bf A.1.2}$$

$$F''(T) = \cases{ 6 U^4 & if $0 < T \le a_\ast$ \cr
                \phantom{x} & \cr
                 2 & if $b_\ast \le T < \infty$. \cr}\leqno{\bf A.1.3}$$

$$H(T) = \left( c_{{}_\ell} - c_{{}_s}\right)\big(F(T) - T F'(T)\big) \leqno{\bf A.1.4}$$


$$H'(T) = -\left( c_{{}_\ell} - c_{{}_s}\right) T\, F''(T)\leqno{\bf A.1.5}$$


$$H(T) = \left( c_{{}_\ell} - c_{{}_s}\right) 
          \cases{ 3U^2 & if $0 < T \le a_\ast$ \cr
                \phantom{x} & \cr
                 - T^2 & if $b_\ast \le T < \infty$. \cr}\leqno{\bf A.1.6}$$

$$H'(T) = \left( c_{{}_\ell} - c_{{}_s}\right) 
          \cases{ -6U^3 & if $0 < T \le a_\ast$ \cr
                \phantom{x} & \cr
                 - 2T & if $b_\ast \le T < \infty$. \cr}\leqno{\bf A.1.7}$$

$$F(T) \le C_\ast\big( U^2 + T^2). \leqno{\bf A.1.8}$$

$$U^2 + T^2 \le C_\ast + F(T). \leqno{\bf A.1.9}$$

$$F''(T) \ge c_\ast +  C_\ast^{-1}U^4.\leqno{\bf A.1.10}$$

$$|H(T)| \le C_\ast \big(U^2 + T^2\big). \leqno{\bf A.1.11}$$

$$|H'(T)| \le C_\ast \big( U^3 + T\big). \leqno{\bf A.1.12}$$

$$\big|F(T_1) - F(T_2)\big| \le C_\ast \big|U_1^2 - U_2^2 \big| + C_\ast \big| T_1 - T_2 \big|
= C_\ast \big| T_1 - T_2 \big| \, \bigg(U_1U_2^2 + U_1^2U_2 + T_1 + T_2 \bigg) .
\leqno{\bf A.1.13}$$

$$\big|H(T_1) - H(T_2)\big| \le C_\ast \big|U_1^2 - U_2^2 \big| + C_\ast \big| T_1 - T_2 \big|
= C_\ast \big| T_1 - T_2 \big| \, \bigg(U_1U_2^2 + U_1^2U_2 + T_1 + T_2 \bigg) .
\leqno{\bf A.1.14}$$
$$T_2 - T_1 \le c_\ast^{-1} \bigg(F'\big(T_2\big)
- F'\big(T_1\big)\bigg) \leqno{\bf A.1.15}.$$
$$U_1^3 - U_2^3 \le C_\ast \bigg(F'\big(T_2\big)
- F'\big(T_1\big)\bigg) \leqno{\bf A.1.16}.$$
\bigskip
PROOF.  The first twelve assertions above are either definitions from Section~2.9
or follow immediately from these definitions.  The rest of the
proof is in three steps.

{\bf Step 1.}  Suppose $h \colon \R^+ \to \R$ is a continuously differentiable
decreasing function such that
$$h(T) = \cases{ A T^{-2} & if $0 < T \le a_\ast$ \cr
                \phantom{x} & \cr
                 -B T^2 & if $b_\ast \le T < \infty$. \cr}$$
for positive constants $A$ and $B$. Then there is a number $c$ (depending only
on $h$) such that
$$\big|h(T_1) - h(T_2)\big| \le c \big|T_1^{-2} - T_2^{-2} \big| + c \big| T_1 - T_2 \big|
= c \big| T_1 - T_2 \big| \, \bigg(T_1^{-1}T_2^{-2} + T_1^{-2}T_2^{-1} + T_1 + T_2 \bigg).$$
\smallskip
{\bf Proof.}  According to the Mean Value Theorem there is a number $T_1 < S < T_2$ such
that
$$\big|h(T_1) - h(T_2)\big| =  \big| T_1 - T_2 \big| \, h'(S)$$
We consider several cases.
\smallskip
Case 1.  Suppose $b_\ast \le S$.  Then $\big|h'(S)\big| = 2 B S \le 2 B T_2$
so that any $c$ not less than $2B$ will do.
\smallskip
Case 2.  Suppose $a_\ast / 2 \le S \le b_\ast$.  If $D$ is a bound for
$|h'(s)|$ for $a_\ast /2 \le s \le b_\ast$ then $h'(S) \le (2D/a_\ast)T_2$
so that any $c$ not less than $(2D/a_\ast)$ will do.
\smallskip
Case 3.  Suppose $S \le a_\ast$ and $T_2 \le a_\ast$.  Then
$h(T_1) - h(T_2) = A \big( T_1^{-2} - T_2^{-2} \big)$ so that an $c$ not less
than $A$ will do.
\smallskip
Case 4.  Suppose $S \le a_\ast$ and $T_2 \ge a_\ast$.  Then
$$h(T_1) 
= {A \over T_1^2} = {4A \over 3} \left({1 \over T_1^2} -{1 \over \left(2 T_1^2\right)}\right) 
\ge {4 A\over 3} \left({1 \over T_1^2} -{1 \over \left( T_2^2\right)}\right)$$
and, since $T_2 \ge a_\ast$ there is some constant $c_1$ such that
$h(T_2) \le c_1 T_2^2 \le (4 c_1 / 3) \big(T_2^2 - T_1^2\big)$.  Hence any $c$ not
less than both
$4A/3$ and $4c_1/3$ will do.
\smallskip
Assertion A.1.14 now follows immediately from the estimate by setting $h(T) = H(T)$.
To verify assertion A.1.13 we set
$$h(T) = \cases{ F(T)  & if $0 < T \le T_\star$ \cr
                \phantom{x} & \cr
                 -F(T) & if $T_\star \le T < \infty$ \cr}$$
and check that
$\big|F(T_1) - F(T_2)\big| \le \big|h(T_1) - h(T_2)\big|.$
\bigskip
{\bf Step 2.}  We use the Mean Value Theorem together with
A.1.10 to infer the existence of $T_1 < T^\ast < T_2$ with
$$F'\big(T_2\big) - F'\big(T_1\big) = F''\big(T^\ast\big)
\big(T_2 - T_1\big) \ge c_\ast \big(T_2 - T_1\big)$$
which gives A.1.15
\bigskip
{\bf Step 3}  In order to  prove assertion A.1.16 we define
$a - 2^{-1/3} a_\ast$ so that
$${1 \over a^3} - {1 \over a_\ast^3} = { 1\over a_\ast^3}$$
and consider three  cases.
\smallskip
Case 1.  If $0 < T_1 < T_2 \le a_\ast$ then
$${1\over T_1^3} - {1 \over T_2^3} = {1 \over 2} \bigg(
F'\big(T_2 \big) - F'\big(T_1\big) \bigg)$$
from which A.1.16 follows in case $C_\ast \ge 1/2$.
\smallskip
Case 2. If $0 < a \le  T_1 < T_2 < \infty$ then
$${1 \over T_1^3} - {1 \over T_2^3}
=\big(T_2 - T_1\big){\big(T_2^2 + T_2T_1 + T_1^2\big)
\over T_1^3T_2^3} 
< 3 {\big(T_2 - T_1\big) \over T_1^3 T_2} < {3 \over
a^4}\big(T_2 - T_1\big)$$
from which A.1.16 follows if $C_\ast \ge 3/a^4$.
\smallskip
Case 3. If $0 < T_1 < a < a_\ast < T_2 < \infty$ then
$$\eqalign{
{1 \over T_1^3} - {1 \over T_2^3} 
&
=
{1 \over T_1^3} - {1 \over a_\ast^3}+{1 \over a_\ast^3} - {1 \over T_2^3} 
\cr &
<
{1 \over T_1^3} - {1 \over a_\ast^3} +  {1 \over a_\ast^3} 
\cr &
\le 2 \left( {1 \over T_1^3} - {1 \over a_\ast^3}\right)
\cr &
= F'\big(a_\ast\big) - F'\big(T_1\big)
\cr &
< F'\big(T_2\big) - F'\big(T_1\big)
\cr
}$$
from  which A.1.16 follows in case $C_\ast \ge 1$.  \E
\bigskip
{\bf A.2 Proposition (Bulk energy functions whose integrals
decrease when heat flows from hot to cold).}
{\sl 
Suppose $G_s, \, G_\ell \colon \R^+ \to \R^+$ are continuously
differentiable functions and $\l K \r$ is a crystal in $\KK$.
Then the following two conditions are equivalent.
$$\int_K G_s(T+\Delta T)\, d \LL
+\int_{\O \sim K} G_\ell(T+\Delta T)\, d \LL
\le\int_K G_s(T)\, d \LL
+\int_{\O \sim K} G_\ell(T)\, d \LL\leqno{(1)}$$
whenever $T$ and $T+\Delta T$ are temperature fields with
$$\sup\big\{ T(x) + \Delta T(x) \colon \Delta T(x) > 0 \big\} <
\inf\big\{ T(x) + \Delta T(x) \colon \Delta T(x) < 0 \big\}.$$
In saying the $T$ and $T+\Delta T$ are temperature fields we
mean that $c_KT$ and $c_K(T + \Delta T)$ are heat
distributions in $\QQ$.
\smallskip\noindent
(2) There is a continuously differentiable convex function
$F_\ast\colon \R^+ \to \R^+$ and a constant $B$ such that,
for each $T$,
$$G_s(T) = c_{{}_s} F_\ast(T)\quad\hbox{and}\quad
G_\ell(T) = c_{{}_\ell} F_\ast(T) + B.$$
}
\bigskip
PROOF.  Assume the first condition holds. We infer that
$$dG_s(S) - dG_s(T) \le 0, \,
dG_\ell(S) - dG_\ell(T) \le 0, \,
dG_s(S) - dG_\ell(T) \le 0, \,
dG_\ell(S) - dG_s(T) \le 0$$
whenever $0 < S < T < \infty$.
Since $T= c_K^{-1}Q$ we have
$$dG_s(T)= G_s'(T){dT \over dQ} dQ = {1 \over c_{{}_s}} G_s'(T)dQ,
\quad\hbox{and}\quad
dG_\ell(T)= G_\ell'(T){dT \over dQ} dQ =
{1 \over c_{{}_\ell}}G_\ell'(T)dQ.$$
from which we infer that
$${1 \over c_{{}_s}} G_s'(S)
\le{1 \over c_{{}_s}} G_s'(T), \,
{1 \over c_{{}_\ell}} G_\ell'(S)
\le {1 \over c_{{}_\ell}} G_\ell'(T),\,
{1 \over c_{{}_s}} G_s'(S)
\le{1 \over c_{{}_\ell}} G_\ell'(T),\,
{1 \over c_{{}_\ell}} G_\ell'(S)
\le{1 \over c_{{}_s}} G_s'(T).$$
It follows that
${1 \over c_{{}_s}} G_s'$ and
${1 \over c_{{}_\ell}} G_\ell'$
are convex functions whose difference is constant.  We infer that
the first condition
implies the second.  The other implication is left to  the
reader. \E
\bigskip
{\bf A.3 Proposition (The Gibbs-Thomson function $H$ effectively
determines the bulk energy function $F$).}
{\sl
Suppose $H_\ast\colon \R^+ \to \R$ is continuous and
$$0 < c_0 < \infty, \quad
0 < T_0< \infty, \quad -\infty < F_0 < \infty.$$
Then there is a unique continuously differentiable function
$F_\ast \colon \R^+ \to \R$ with $F_\ast(T_0) = F_0$ such that
$$H_\ast(T) = c \big(F_\ast(T) - T F_\ast'(T)\big).$$
If $H_\ast$ is continuously differentiable and
nonincreasing, then $F_\ast$ is convex.
}
\bigskip
PROOF. Our differential equation above can be rewritten
$$F_\ast'(T) - {1 \over T} F_\ast(T) = {-c \over T}H_\ast(T)$$
which is a standard first order ordinary differential equation
for which existence and uniqueness of solutions is well known
for arbitrary initial conditions as set forth on page~11 of~[BR].
If $H_\ast$ is continuously differentiable we compute
$$\eqalign{F_\ast''(T)
&= {-1\over T^2} F_\ast(T)+ {1 \over T} F_\ast'(T)
   + {1 \over T^2}H_\ast(T) - {1 \over T}H_\ast'(T)T
\cr &
= {-1\over T^2} F_\ast(T)+ {1 \over T}\Big(
 {1 \over T} F_\ast(T) + {-c \over T}H_\ast(T)\Big)
   + {1 \over T^2}H_\ast(T) - {1 \over T}H_\ast'(T)T
\cr &
= {-1 \over T} H_\ast'(T)\cr}$$
so that $F_\ast''(T)$ is nonnegative in case $H_\ast$ is
nonincreasing. \E
\bigskip
\centerline{\larger Appendix B.  Measure and function spaces}
\bigskip
{\bf B.1 A metric space of Radon measures.}
We denote by $\MM$ the vector space of all signed Radon measures
(Daniell integrals
[F 2.5]) $\mu$ on $\O$ such that $$\mu(\O) = 0.$$
Associated with such $\mu$'s are its {\bf mass norm}
$$|\mu| = \sup \left\{ \int_\O \varphi \, d \mu \colon \varphi \hbox{ is 
smooth with } \sup|\varphi| \le 1 \right\}$$
and its {\bf Monge-Kantorovich norm}
$$\big\| \mu \big\|_\ast 
= \sup \left\{ \int_\O \varphi \, d \mu \colon \lip (\varphi) \le 1 \right\}.$$
{\sl For each $M < \infty$ the set
$$\MM \cap \big\{ \mu \colon |\mu| \le M \big\}$$
is compact in the $\| \cdot \|_\ast$ topology.
}
\bigskip
{\bf B.2  The vector space $\L_1(\O)$.}
We denote by ${\L}_1(\O)$ the real vector space of all ${\cal L}^n$ summable functions 
$f  \colon \O \to \R$.
The $\L_1$ norm of such an $f$ is the number
$$\| f \|_{{}_{{\bf L}_1}} = \int_\O |f| \,d {\cal L}^n .$$
If $f$ belongs to $\L_1(\O)$ with $\int_\O f \, d \LL = 0$ then the measure 
$\LL \wedge f$  belongs to $\MM$ and one abbreviates.
$$\| f \|_\ast = \left\|\LL \wedge f\right\|_\ast.$$
If $g$ and $h$ belong to $\L_1(\O)$ with
$\int_\O g \, d \LL = \int_\O h \, d \LL$ then the number 
$\| g - h\|_\ast$ gives a the {\bf Monge-Kantorovich distance}
between $g$ and $h$.
\bigskip
{\bf B.3 The vector spaces $\L_p(\O)$.}  For $p > 1$ we denote
by $\L_p(\O)$ the vector subspace of $\L_1(\O)$ consisting of
$f$'s whose absolute values
are $p$-th power summable.  The $\L_p$ norm of such an
$f$ is the number
$$\| f \|_{{}_{{\bf L}_p}}
= \left(\int_\O |f|^p \,d {\cal L}^n \right)^{1 \over p}.$$
Equivalently we can set $q = p/(p-1)$ (so that $1/p + 1/q =
1$) and realize
$\| f \|_{{}_{{\bf L}_p}}$
as the supremum of the numbers
$$\int_\O \varphi f\,d {\cal L}^n$$ 
corresponding to all $\varphi$'s in $\L_q(\O)$ with
$\| \varphi \|_{{}_{{\bf L}_q}} = 1$ [FH2.4.16].
With this norm $\L_p(\O)$ is a Banach space.

Suppose $f, \, f_1, \, f_2, \, f_3, \, \ldots \in \L_p(\O)$.
We say that the $f_i$'s {\bf converge strongly to }$f$, written
$f_i \to f$, as $i \to \infty$ if and only if
$$\|f_i - f \|_{{}_{{\bf L}_p}}\to 0
\quad\hbox{as } i \to \infty.$$
We say
that the $f_i$'s {\bf converge weakly to }$f$, written
$f_i \h f$, as $i \to \infty$ if and only if
$$\int_\O \varphi \big(f_i -  f\big)\,d {\cal L}^n \to 0
\quad\hbox{as } i \to \infty$$
for each (fixed) function $\varphi$ in $\L_q(\O)$;
it follows that
$$ \| f \|_{{}_{{\bf L}_p}} \le \lim\inf_i
 \| f_i \|_{{}_{{\bf L}_p}}.$$

Suppose $f, \, f_1, \, f_2, \, f_3, \, \ldots \in \L_p(\O)$
with $\sup_i \| f_i \|_{{}_{{\bf L}_p}} < \infty$ and that
$f_i \h f$ as $i \to \infty$ (weak convergence in $\L_p(\O)$).
Suppose also that $\varphi, \, \varphi_1, \, \varphi_2, \,
\varphi_3, \, \ldots \in \L_q(\O)$ and that
$\varphi_i \to \varphi$ as $i \to \infty$ (strong convergence
in $\L_q(\O)$).  Then
one uses H\"older's inequality [FH 2.4.14] to check
$$\int_\O \varphi_i\,f_i\,d {\cal L}^n \to 
\int_\O \varphi\, f \,d {\cal L}^n 
\quad\hbox{as } i \to \infty.$$
One can then combine this estimate with  Ascoli's theorem to
infer additionally that
$$\|f_i - f\|_\ast \to 0
\quad\hbox{as } i \to \infty.$$
Whenever $ f_1, \, f_2, \, f_3, \, \ldots \in \L_p(\O)$ with 
$\sup_i \| f_i \|_{{}_{{\bf L}_p}} < \infty$ then there
will always be
some subsequence $i(1), \, i(2), \, i(3), \, \ldots$ of 
$1, \, 2, \,  3, \, \ldots$ and $f \in \L_p(\O)$ such
that $f_{j(i)}$'s converge to $f$ weakly in
$\L_p(\O)$ as $i \to \infty$  [GT Theorem 5.12][YK][MC Theorem
3.2.4(e)].
\bigskip
{\bf B.4 Proposition (Lower semicontinuity of convex
integrals)}.
{\sl
Suppose
\smallskip\noindent
(a) $p, \, q > 1$ with $1/p + 1/q = 1$ and $0 < M < \infty$,
and
\smallskip\noindent
(b) $G\colon(0, \, \infty) \to \R^+$ is a convex function
with $\lim_{t \downarrow 0} G(t) = \lim_{t \to \infty} G(t) = \infty$. 
\smallskip\noindent
The convexity of $G$ implies the existence of
nondecreasing right hand derivates 
$H\colon (0, \, \infty) \to \R$ with
$G(y) - G(z) \ge H(z)(y-z)$ whenever $y, \, z \in (0, \, \infty)$.
Now suppose additionally
\smallskip\noindent
(c) $H(z) \le Mz^{p-1}$ for each $z > M$;
\smallskip\noindent
(d) $f, \, f_1, \, f_2, \, f_3, \, \ldots\in \L_p(\O)$ such that,
for each $i$<
$$\|f_i\|_{{}_{{\bf L}_p}}
\le M \quad\hbox{and}\quad
\int_\O G\left(f_i\right)\, d \LL \le M,$$
and $f_i \h f$ as $i \to \infty$ in $\L_p(\O)$;
\smallskip\noindent
(g) $g, \, g_1, \, g_2, \, g_3, \, \ldots\in \L_1(\O)$ such
that, for each $i$ and each $x$,
$0 \le g_i(x) \le M$, and
$\|g_i - g \|_{{}_{{\bf L}_1}} \to 0$ as $i \to \infty$.
\smallskip\noindent
Then
\smallskip\noindent
$$\int_\O g\,G\left(f\right)\, d \LL \le 
\lim\inf_i
\int_\O g_i\, G\left(f_i\right)\, d \LL.$$
}
\bigskip
PROOF.  By contradiction.
The negation of our assertion implies the existence of
$\epsilon > 0$ and a subsequence of $1,\, 2, \,3, \ldots$
(which, for notational simplicity, we assume is our
original sequence $1, \, 2, \,3, \, \ldots$) such that
$$\int_\O g\,G\left(f\right)\, d \LL \ge 
\int_\O g_i\, G\left(f_i\right)\, d \LL - 2 \epsilon$$
for each $i$.
The convexity of $G$ together with its divergence to $\infty$
at $0$ and $\infty$ imply
that $G$ assumes its minimum value at some point $0 < z <
\infty$.  We fix such $z$ and for $t \ge 0$ and $0 < x <\infty$ we set
$$h(t, \, x) = z + (1-t)(f(x) - z) = tz + (1-t)f(x),$$
$$ h_i(t, \, x) = z + (1-t)(f_i(x) - z) = tz + (1-t)f_i(x).$$
Since we have used only translations and rescalings in the
range in constructing our $h_i(t, \, \cdot)$'s and  $h(t, \,
\cdot)$ we infer, 
for each $t \ge 0$ that $h_i(t, \, \cdot) \h h(t,
\, \cdot)$ in $\L_p(\O)$
as $i \to \infty$.  We check also, for each
$0 < x < \infty$, that $G(h(t, \, x)) \uparrow G(h(0, \, x))$
as $t \downarrow 0$.  We then use Lebesgue's increasing
convergence theorem [FH 2.4.7] to conclude the existence
of $t > 0$  such  that
$$\eqalign{\int_{x \in \O} g(x)\,G\left(h(t, \, x)\right)\, d \LL x
&\ge 
\int_{x \in \O} g_i(x)\, G\left(f_i(x)\right)\, d \LL x - \epsilon
\cr &\ge
\int_{x \in\O} g(x_i)\, G\left(h_i(t, \, x)\right)\, d \LL x - 
\epsilon}\leqno{(\ast)}$$
for each $i$.  We now estimate
$$\eqalign{\int_{x \in\O} g_i(x)\, G\left(h_i(t, \, x)\right)\,
d \LL x\,-
& \int_{x \in\O} g(x)\,G\left(h(t, \, x)\right)\, d \LL x\cr
& =
\int_{x  \in\O} g_i(x)\,\big( G\left(h_i(t, \, x)\right) -
G\left(h(t, \,x)\right)\big)\, d \LL x \cr
&\phantom{X}
+ \int_{x  \in\O} \big(g_i(x)- g(x)\big) G\left(h(t, \, x)\right)
 \, d \LL x \cr
&\ge
\int_{x \in\O} g_i(x)\, H(h(t, \, x))\big(h_i(t, \, x) - h(t, \, x)\big)
\, d \LL x .\cr
&\phantom{X}
+ \int_{x  \in\O} \big(g_i(x)- g(x)\big) G\left(h(t, \, x)\right)
 \, d \LL x \cr
}\leqno{(\ast\ast)}$$
We combine assumption~(c) with the fact that $h(t, \, x) \ge tz$
for each $x$ to conclude that the functions
$x \mapsto g_i(x)H(h(t, \, x))$ belong to $\L_q(\O)$
with uniformly bounded $\L_q$ norms and that they converge
strongly in $\L_q(\O)$ to
$x \mapsto g(x)H(h(t, \, x))$.
The weak convergence of the $h_i(t, \, \cdot)$'s to $h(t, \,
\cdot)$ then implies that  the first integral in the final
expression in $(\ast\ast)$
converges to zero as $i \to \infty$ as in B.3.  We infer further
from assumption~(c)
and the fact that $h(t, \, t)\ge tz$ for each $x$
that the function $G(h)$ is $\LL$ summable.
We infer from Lebesgue's bounded convergence theorem  that the
second integral in the final
expression in $(\ast\ast)$
also converges to zero as $i \to \infty$. 
These two estimates are incompatible
with $(\ast)$.  This completes our proof by contradiction
of our assertion. \E
\bigskip
{\bf B.5 Remark.}  Results analogous to that of Proposition~B.4
hold in case the domain of the convex function $G$ is a
general interval $(a, \, b)$ which also contains  the images
of the $f_i$'s.  Additionally the assumption analogous to~(c)
can be relaxed provided that at endpoints at which $G$ does
not diverge to~$\infty$ one extends $G$ continuously to
include that endpoint in its domain.
\bigskip
{\bf B.6 Corollary (A compactness estimate).}
{\sl
Suppose $L$ and $M$ are positive numbers and ${\cal P}(L, \, M)$ is the
space of all nonnegative functions $f$ for which
\smallskip\noindent
(a) $f \in \L_2(\O)$ with $\int_\O f \, d \LL = L$ and $\| f \|_{{}_{{\bf L}_2}} \le M$.
\smallskip\noindent
(b) $1/f \in \L_2(\O)$ with $\| 1/f \|_{{}_{{\bf L}_2}} \le M$.
\smallskip\noindent
Then ${\cal P}(L, \, M)$ is compact in the topology of the
Monge-Kantorovich distance $\| \cdot \|_\ast$.
}
\bigskip
{\bf B.7 The vector space  $\H^1(\O)$.}
We denote by ${\H}^1(\O)$ the vector subspace of $\L_2(\O)$
of all functions $f$ which have square summable 
partial derivatives in the sense of distributions, 
i.e. for which there exist functions $g_1, \, \ldots, \, g_n$ in $\L_2(\O)$ such that
$$\int_{\O} f{\partial \varphi \over \partial x_i} \, d {\cal L}^n = 
\int_{\O} - g_i \varphi \, d {\cal L}^n$$
for each $i$ and for each smooth function $\varphi \colon \O \to \R$.
For such $f$ and $g_1, \, \ldots, \, g_n$ 
we abbreviate $\nabla f = \left(g_1, \, g_2, \, \ldots, \,
g_n\right) \colon \O \to \R^n$, and for each
$\LL$ measurable subset $A$ of
$\O$ we denote
$$\dir(f;\, A) = \int_{A} \left( g_1^2 + \ldots + g_n^2\right)\,
d {\cal L}^n
= \int_{A} \left| \nabla f \right|^2 \, d {\cal L}^n$$
as the usual Dirichlet's integral of $f$.  We abbreviate
$$ \dir (f) = \dir(f; \,\O).$$

If  function $f$ belongs to $\H^1(\O)$ then the usual
Poincar\'{e}'s inequality for the cube $\big[0, \, \rho_\ast\big]^n$
guarantees that
$$\int_\O f^2 \, d \LL \le {1 \over \rho_\ast^n}\left(\int_\O f\, d
\LL\right)^2 + \rho_\ast^2\dir (f).$$

Suppose $f_1,\,  f_2, \, f_3, \, \ldots \in {\H}^1(\O)$ with
$\infty > \sup_i \| f_i \|_{{}_{{\bf L}_2}} + \sup_i \dir\big(f_i\big)$.
Then there is a subsequence $j(1), \, j(2), \, j(3), \, \ldots$ of
$1, \, 2, \, 3, \, \ldots$ and $f \in {\H}^1(\O)$ such that
$$\lim_{i \to \infty} 
\| f - f_{j(i)} \|_{{}_{{\bf L}_2}} = 0,$$
$$\nabla f_{j(i)} \h \nabla f \hbox{ as } i \to \infty \quad
\left(\hbox{weak
convergence in } \L_2\left(\O, \, \R^n\right)\right),$$
and hence
$$\dir(f) \le \lim\inf_i \dir\left(f_{j(i)}\right).$$
[MC Theorem 3.4.4, Rellich's theorem].
\bigskip
{\bf B.8 Proposition (Functions in $\H^1(\O)$ have small
integrals over small sets).}
{\sl
Suppose $A$ is an $\LL$ measurable subset  of  $\O$ and
$f$ in $\H^1(\O)$ has average value
$$AV(f) = {1 \over |\O|} \int_\O f \, d \LL.$$
(1)  If $n \ge 3$ then
$$\int_A f^2 \, d \LL \le C\, \LL(A)^{2 \over n}\, \dir(f)
+ 2 \LL(A)\, AV(f)^2;$$
here $C$ is a constant depending only on $n$ and $\O$.
\smallskip\noindent
(2) If $n = 2$ and $0 < \epsilon < 1$ then
$$\int_A f^2 \, d \LL \le C_\epsilon\, \LL(A)^{1-\epsilon} \, \dir(f)
+ 2 \LL(A)\, AV(f)^2;$$
here $C_\epsilon$ is a constant depending only on $\epsilon$ and $\O$.
}
\bigskip
PROOF.  We will combine various H\"older's
inequalities with the generalized Poincar\'e type inequality
given in [GT Theorem 7.10, page 148] which says, in
particular, for suitable functions $u \colon \R^n \to \R$
having compact support and numbers $1 < p < n$ that
there is a constant $C$ such that
$$\| u \|_{{}_{{\bf L}_{np/(n-2)}}} \le C \| Du \|_{{}_{{\bf
L}_p}}.$$
Here and below we denote by ``C'' a general finite constant which
need not have the same value in different equations.
\bigskip
{\bf Step 1.} We set $g = f - AV(f)$ so that
$$\int_A f^2 \,  d \LL \le 2 \int_A g^2 \, d \LL + 2 \LL(A)
\, AV(f)^2,$$
and, by Poincar\'e's inequality,
$$\int_\O g^2 \, d \LL \le \rho_\ast^2  \,\dir(g) = \rho_\ast^2
\, \dir(f).$$
\bigskip
{\bf Step 2.} We denote by $\pi \colon \R^n \to \O$ the
simply connected locally isometric
covering mapping of our torus $\O$ and define $h = g \circ \pi
\colon \R^n \to \R$ and $B = \pi^{-1}(A) \cap [0, \,
\rho_\ast]^n$.  We fix a Lipschitz function $\varphi \colon
\R^n \to [0, \, 1]$ with $\lip(\varphi) = 1/\rho_\ast$ and
$$\varphi (x) = \cases{ 1 & in $x \in [0, \, \rho_\ast]^n$\cr
\phantom{x}& \cr
0 &if $\d \left(x, [0, \, \rho_\ast]^n\right) \ge \rho$\cr}$$
and set $k = \varphi h$ so that
$|k| \le |h|$ and
$\nabla k = \varphi \nabla h + h \nabla \varphi$ and hence
$$\int_{{\bf R}^n} k^2 \, d \LL \le
\int_{[-\rho_\ast, \, 2\rho_\ast]^n} h^2 \, d \LL = 3^n
\int_\O g^2 \, d \LL,$$
$$\eqalign{\int_{{\bf R}^n} |\nabla h|^2 \, d \LL 
&\le
2\int_{[\rho_\ast, \, 2\rho_\ast]^n}\left( |\nabla h|^2 + {1 \over
\rho_\ast^2} |h|^2\right) \, d \LL 
\cr
&=
2 \cdot 3^n\left[ \dir(f) + {1 \over \rho^2_\ast}\int_\O g^2 \,
d \LL\right]
\cr
&\le 4 \cdot 3^n \dir(f)\cr}$$
by Step 1.
\bigskip
{\bf Step 3.}  If $n \ge 3$ we set 
$$ p = 2, \qquad P = {np \over n-p}={2n \over n-2},
\qquad R = {P \over 2} = {n\over n-2},
\qquad Q = {n \over 2}$$
and use H\"older's inequality  and [GT Theorem 7.10] to
estimate
$$\eqalign{\int_A g^2 d \LL
&=
\int_B h^2 \, d \LL 
\cr
&\le \LL(B)^{1 \over Q} \,
\|h^2\|_{{}_{{\bf L}_R}}
\cr
&= \LL(A)^{2\over n} \,
\|h\|_{{}_{{\bf L}_P}}^2
\cr
&\le C \, \LL(A)^{2\over n} \,
\|\nabla h\|_{{}_{{\bf L}_2}}^2
\cr
&\le C  \,\LL(A)^{2\over n}\, 4 \cdot 3^n  \,\dir(f)\cr}$$
by Step 2.  Our first conclusion follows.
\bigskip
{\bf Step 4.} If $n = 2$  and $0 < \delta < 1$ we set 
$$ p = 2- \delta, \quad P = {np \over n-p}={2(2 - \delta) \over \delta},
\quad R = {P \over 2} = {2- \delta \over \delta},
\quad Q = {2 - \delta \over 2 - 2 \delta}$$
and use H\"older's inequality twice and [GT Theorem 7.10] to
estimate
$$\eqalign{\int_A g^2 d \LL
&=
\int_B h^2 \, d \LL \cr
&\le \LL(B)^{1 \over Q} \,
\|h^2\|_{{}_{{\bf L}_R}}
\cr
&= \LL(A)^{2- 2 \delta \over 2 - \delta} \,
\|h\|_{{}_{{\bf L}_P}}^2
\cr
&\le C \LL(A)^{2- 2 \delta \over 2 - \delta} \,
\|\nabla h\|_{{}_{{\bf L}_{2-\epsilon}}}^2
\cr
&\le C \LL(A)^{1 - \epsilon} \, 4 \cdot 3^n\, \dir(f)}$$
for $1 - \epsilon = (2 - 2 \delta)/(2 - \delta)$ by Step 2.
Our  second conclusion follows.
\bigskip
{\bf B.9 Proposition (A domination estimate).}
{\sl
Corresponding to each $\epsilon > 0$ there exists $M < \infty$ such that
$$\|f \|_{{}_{{\bf L}_2}}
\le M \|f\|_\ast + \epsilon \dir(f)$$
whenever $f$ belongs to $\H^1(\O)$ with 
$\int_\O f \, d \LL = 0$.  
}
\medskip
PROOF.  By contradiction.
The negation of the assertion implies the existence of some
$\epsilon > 0$ (which we fix)
such that for each $M$ there will exists some $f$ with 
$\|f \|_{{}_{{\bf L}_2}} > M \|f\|_\ast + \epsilon \dir(f)$
or, equivalently,
$(1 + \epsilon)\|f \|_{{}_{{\bf L}_2}} > M \|f\|_\ast + 
\epsilon\left( \|f \|_{{}_{{\bf L}_2}} + \dir(f)\right)$.
Without loss of generality we assume
$ \|f \|_{{}_{{\bf L}_2}} + \dir(f) = 1$; this implies
$ \|f \|_{{}_{{\bf L}_2}} \ge  \epsilon/(1 + \epsilon)$.
We fix $f_i$ to be such an $f$ chosen corresponding to
$M = i$; this implies $\lim_i \|f_i\|_\ast = 0$.
We use B.1 and B.8 to infer the existence of a limit $g$ for
which
$\|g \|_{{}_{{\bf L}_2}} \ge \epsilon/(1 + \epsilon)$ while
$\|g\|_\ast = 0$.  This is not possible.  \E
\bigskip
{\bf B.10 Proposition (Sobolev estimates in dimensions 2 and 3).}
{\sl

\noindent
(1) If the ambient dimension $n$ equals $2$ there are
constants $C_p$ (depending on $1 < p < \infty$
and on $\rho_\ast$)
such that, for each $f$ in $\H^1(\O)$,
$$\left(\int_\O f^p \, d {\cal L}^2\right)^{1 \over p}
\le C_p \left(\int_\O f^2 \, d {\cal L}^2\right)^{1 \over 2}
+ C_p\left( \int_\O |\nabla f|^2 \, d {\cal L}^2\right)^{1 \over 2}.$$
\smallskip
\noindent
(2) If the ambient dimension $n$ equals $3$ there is a
constant $C$ (depending only on $\rho_\ast$) such that,
for each $f$ in $\H^1(\O)$,
$$\left(\int_\O f^6 \, d {\cal L}^3\right)^{1 \over 6}
\le C \left(\int_\O f^2 \, d {\cal L}^3\right)^{1 \over 2}
+ C\left( \int_\O |\nabla f|^2 \, d {\cal L}^3\right)^{1 \over 2}.$$
}
\bigskip
PROOF.
Our torus $\O$ is naturally identified with the quotient of $R^n$
by the lattice generated by $\rho_\ast{\bf e}_i$ $(i = 1,
\, \ldots, \, n)$.  Set $J = [-\rho_\ast, \, 2 \rho_\ast]^n
\subset \R^n$
and let $g \colon J \to \R$ denote the pullback of $f$ to $J$. 
Also set $h(x) = \sup\big\{ 1, \, \rho_\ast^{-1}\hbox{dist}(x, \,
\partial J)\big\}$ for $x$ in $\O$.  We estimate
$$\int_J |\nabla (gh)|^2 \, d \LL
\le C_0\int_\O f^2 \, d \LL +
\le C_0\int_\O|\nabla f|^2 \, d \LL$$
for a suitable constant $C_0$ depending on $n$ and
$\rho_\ast$.  We then apply the Sobolev estimates given in
[GT Theorem~7.10, page 148] to the function $gh$ and then
by restriction (from $J$ to $[0, \, \rho_\ast]^n$) to $f$.
\E
\bigskip
\centerline{\larger Appendix C.  Currents and varifolds and
convergence}
\bigskip
In this appendix we review some terminology and basic facts
about currents and varifolds.  We assume the reader will
consult the discussion in Chapter~3 of the paper~[ATW] as
necessary.
\bigskip
{\bf C.1 Terminology for variations of integrals over currents
and varifolds.}  Suppose $1 \le k \le n$ are fixed integers.
\medskip\noindent
(1) We identify the Grassmann manifold $\G_0(n, \, k)$ 
of all oriented $k$~dimensional vector subspaces of $\R^n$
with the subset of
the Grassmann vector space $\wedge_k\R^n$ consisting of all
simple unit $k$~vectors in $\R^n$.
\medskip\noindent
(2) By a {\sl parametric integrand} we mean a continuous function
$\Psi \colon \R^n \times \G_0(n, \, g)\to \R$.
For our purposes here we call a parametric integrand $\Psi$
{\sl elliptic} provided that for each $x$ there is
a convex function $F \colon \wedge_k\R^n \to \R^+$
satisfying the following conditions:
\smallskip\noindent
(i) $F$ is positively homogeneous of degree~1
and is positive and twice continuously differentiable
on $\wedge_kR^n\sim
\{0\}$;
\smallskip\noindent
(ii) $F$ has positive second derivatives when
restricted to each straight line not containing the origin.
\medskip\noindent
(3) Associated with each smooth 
vector field $g \colon \R^n \to \R^n$  having compact support
are the {\sl deformations}
$G_t \colon  \R^n \to \R^n$, $t \in \R$, given by setting
$$G_t(x) =x + t\, g(x)$$
for each $t$ and $x$.  The initial velocity vector field of
$G_t$ is $g$.
\medskip\noindent
(4) Associated with each smooth parametric integrand
$\Psi \colon \R^n \times \G_0(n, \, k)\to \R$.
and with each smooth initial velocity vector field $g$ is the
function $H(\Psi, \, g) \colon \R^n \times \G_0(n, \, k) \to
\R$ given by setting
$$H(\Psi, \, g)(x, \, \sigma) = {d \over dt}
\Big[|\sigma(t)| \Psi\left( |\sigma(t)|^{-1} \sigma(t) \right)\Big]
\Bigg|_{t = 0}\quad \hbox{where}\quad
\sigma(t) = \big<\sigma, \, \wedge_k DG_t(x)\big>.$$
\medskip\noindent
(5) For our purposes here, a $k$~{\sl current $T$ in}
$\R^n$ is a continuous linear functional on the real vector
space of infinitely differentiable differential $k$~forms $\omega$
in $\R^n$ having compact support which current also
has compact support.
In accordance with [FH 4.1.7, page 357] such a $k$~current~$T$ is 
{\sl representable by integration} when there is a Radon measure
$\|T\|$ over $\R^n$ having compact support
and a $\|T\|$ measurable $\wedge_k\R^n$
valued function $\vec T$ of unit length such that
$$T(\omega) = \big( \|T\| \wedge \vec T\big)(\omega)=
\int_{{\bf R}^n}\left< \vec T, \, \omega\right> \, d \|T\|.$$
A $k$~current $T$ is called (real) {\sl rectifiable} when there is
a $k$~rectifiable set $S$, a ${\cal H}^k$ summable positive
density function $\theta \colon S \to \R^+$
and an associated oriented Borel tangent plane valued function
$\sigma\colon S \to \G_0(n, \, k)$ such that
$$T(\omega) = {\bf t}\big(S, \, \theta, \, \sigma\big)(\omega)
= \int_S \left< \sigma, \, \omega\right> \, \theta \, d
{\cal H}^k, $$
in particular, $\|T\| = {\cal H}^k \rightangle S \wedge \theta$ and
$\vec T = \sigma$.

In accordance with [FH  4.1.7, page
359], each smooth mapping $f\colon \R^n \to
\R^n$ naturally maps a $k$~currents $T$ in $\R^n$ to
a $k$~currents $f_\sharp T$ in $\R^n$.

Associated with each (real) rectifiable $k$~current $T = {\bf t}(S, \,
\theta, \, \sigma)$ and each parametric integrand 
$\Psi \colon \R^n \times \G_0(n, \, g)\to \R$ is the {\sl integral
$\Psi(T)$ of $\Psi$ over $T$} given by setting
$$\Psi(T) = \int_{x \in S} \Psi(x, \, \sigma(x))\, \theta(x)\,
d {\cal H}^kx.$$
\medskip\noindent
(6) By an {\sl oriented $k$~varifold in}
$\R^n$ we mean a Radon measure over $\R^n \times \G_0(n, \, k)$
having compact support.
Suppose $V, \, V_1, \, V_2, \, V_3, \ldots$ are $k$ varifolds
in $\R^n$.  We say that $V_j$ {\sl converges to $V$ weakly} 
as $j \to \infty$ and write $V_j \h V$ as $j \to
\infty$ if and only if
$$\int_{{\bf R}^n \times {\bf G}_0(n, \, k)} \varphi \, d V_j
\to\int_{{\bf R}^n \times {\bf G}_0(n, \, k)} \varphi \, d V$$
as $j \to \infty$ for each continuous function 
$\varphi \colon \R^n \times \G_0(n, \, k) \to \R$ having
compact support.
Associated with each rectifiable $k$ current ${\bf t}\big(S,
\, \theta, \, \sigma)$ as above is the {\sl rectifiable
oriented $k$ varifold } ${\bf v}\big(S, \, \theta, \,
\sigma\big)$ defined by requiring for each continuous function
$\varphi \colon \R^n \times \G_0(n, \, k)\to \R$ having
compact support that
$$\int_{{\bf R}^n \times {\bf G}_0(n, \, k)}\varphi \, d {\bf v}(S,
\, \theta, \, \sigma) = \int_{x \in S} \varphi\big(x, \,
\sigma(x) \big) \, \theta(x) \, d {\cal H}^k x.$$

By a direct extension of the formula given in [AW 3.2, page~430]
each smooth mapping $f\colon \R^n \to
\R^n$ naturally maps a $k$~varifold $V$ in $\R^n$ to
a $k$~varifold $f_\sharp T$ in $\R^n$.

Associated with each $k$~varifold $V$ and each parametric integrand 
$\Psi \colon \R^n \times \G_0(n, \, g)\to \R$ is the {\sl integral
$\Psi(V)$ of $\Psi$ over $V$} given by setting
$$\Psi(V) = \int_{{\bf R}^n \times {\bf G}_0(n, \, k)} \Psi
d V.$$
\bigskip
{\bf C.2 Theorem (Computing first variations).}
{\sl 
Suppose $\Psi \colon \R^n \times \G_0(n, \, g)\to \R$ is a smooth
parametric integrand and that $g \colon \R^n \to
\R^n$ is a smooth initial velocity vector field
in~$\R^n$ having compact support and $G_t$ gives the associated
deformations as in~C.1(3) above.  Let $H(\Psi, \, g) \colon \R^n \times
\G_0(n, \, k) \to \R$ be the function given in C.1(4) above.
Then
\smallskip\noindent
(1) For each $k$~current $T = {\bf t}\big(S, \, \theta, \,
\sigma\big)$ as above
$${d \over dt}\left( \Psi\big(
G_{t\sharp}(T)\big)\right)\Big|_{t=0}
=\int_{x \in S} H(\Psi, \, g)\big(x, \, \sigma(x)\big)\,\theta(x)
\,d {\cal H}^kx.$$
\smallskip\noindent
(2) For each $k$~varifold $V$ as above
$${d \over dt}\left( \Psi\big(
G_{t\sharp}(V)\big)\right)\Big|_{t=0}
=\int_{(x, \, \sigma)\in {\bf R}^n \times {\bf G}_0(n, \, k)}
H(\Psi, \, g)\big(x, \, \sigma\big)\, d V( x, \, \sigma).$$
\smallskip\noindent
(3) Suppose 
$$T= {\bf t}\big(S, \, \theta, \, \sigma), \,
T_1= {\bf t}\big(S_1, \, \theta_1, \, \sigma_1), \,
T_2= {\bf t}\big(S_2, \, \theta_2, \, \sigma_2), \,
T_3= {\bf t}\big(S_3, \, \theta_3, \, \sigma_3), \,
\ldots$$
are $k$~currents in $\R^n$ associated with the
varifolds
$$V= {\bf v}\big(S, \, \theta, \, \sigma), \,
V_1= {\bf v}\big(S_1, \, \theta_1, \, \sigma_1), \,
V_2= {\bf v}\big(S_2, \, \theta_2, \, \sigma_2), \,
V_3= {\bf v}\big(S_3, \, \theta_3, \, \sigma_3), \,
\ldots.$$
In order that
$$\lim_{j\to \infty} {d \over dt}\left( \Psi\big(
G_{t\sharp}(T_j)\big)\right)\Big|_{t=0}
\to
{d \over dt}\left( \Psi\big(
G_{t\sharp}(T)\big)\right)\Big|_{t=0}$$
as $j \to \infty$ it is sufficient that $V_j  \h V$ as
$j \to \infty$.
}
\bigskip
PROOF.  The proof is left to the reader.
\bigskip
{\bf C.3 Theorem (Current and varifold convergence).}
{\sl
Suppose 
$$T= {\bf t}\big(S, \, \theta, \, \sigma), \,
T_1= {\bf t}\big(S_1, \, \theta_1, \, \sigma_1), \,
T_2= {\bf t}\big(S_2, \, \theta_2, \, \sigma_2), \,
T_3= {\bf t}\big(S_3, \, \theta_3, \, \sigma_3), \,
\ldots$$
are $k$~currents in $\R^n$ associated with the
varifolds
$$V= {\bf v}\big(S, \, \theta, \, \sigma), \,
V_1= {\bf v}\big(S_1, \, \theta_1, \, \sigma_1), \,
V_2= {\bf v}\big(S_2, \, \theta_2, \, \sigma_2), \,
V_3= {\bf v}\big(S_3, \, \theta_3, \, \sigma_3), \,
\ldots.$$
In order that that $V_j \h V$ as $j \to \infty$, it is
sufficient that the following  conditions hold:
\smallskip\noindent
(a) the union of the supports of the $T_j$'s be
bounded;
\smallskip\noindent
(b) the masses $\|T_j\|\R^n$ be bounded;
\smallskip\noindent
(c) $T_j(\omega) \to T(\omega)$ as $j \to \infty$ for each smooth
differential $k$~form;
\smallskip\noindent
(d) there exists an elliptic parametric integrand
$\Psi \colon \R^n \times \G_0(n, \, g)\to \R$ such that
$\Psi(T_j) \to \Psi(T)$ as $j \to \infty$.
}
\bigskip
PROOF.  The proof depends on the two propositions and
corollary following.
\bigskip
{\bf C.4 Proposition (A variant of Jensen's inequality).}
{\sl
Suppose $F, \, G \colon \R^n \to \R^+$ are convex functions
which are twice continuously differentiable on $\R^n\sim \{0\}$.
Suppose also that $F$, $G$ are positively homogeneous of degree~1
and have positive second derivatives when
restricted to each straight line not containing the origin.
In case $n = 1$ we assume additionally that neither $F$ nor
$G$ is everywhere equal to zero.
Then there is a number
$0 < \lambda < \infty$ (depending only on $F$ and $G$)
with the following property.

Suppose $\varphi$ measures $X$ with $ 0 < \varphi(X) < \infty$
and $h \colon X \to \R^n$ is $\varphi$ summable.  Then
the two numbers
$$\int_X (F \circ h)\, d \varphi
-F\left(\int_X  h\, d \varphi\right)\quad
\hbox{and}\quad
\int_X (G \circ h)\, d \varphi
-G\left(\int_X  h\, d \varphi\right)$$
are within factor $\lambda$ of each other above and below.
}
\bigskip
PROOF.
{\bf Step 1.} We assert there is at most one  point $p$ in
$\partial \B^n(0, \, 1)$ with $F(p) = 0$ [resp. with $G(p) =
0$].  The is clear if
$n=1$; we therefore assume $n\ge 2$ and suppose $p \not= q \in
\partial \B^n(0, \, 1)$ with $F(p) = F(q) = 0$.  If $q \not= -p$
then the convexity and positivity of $F$ imply that $F$
vanishes on the line segment $\overline{pq}$ contrary to our
assumption about second derivatives.  If $q = -p$ we choose
$z \in \partial \B^n(0, \, 1)$ with $z \bullet p = 0$ and
estimate
$$F\left({p+z\over 2}\right) < {F(z)\over 2}\quad\hbox{and}\quad
F\left({q+z\over 2}\right) < {F(z)\over 2}$$
(by Taylor's formula and our assumption about second
derivatives applied to the line segments $\overline{pz}$ and
$\overline{qz}$) and
$$F \left({z\over 2}\right)
= F\left({{p+z\over 2} + {q+z\over 2} \over 2}\right)
< {{F(z)\over 2} + {F(z)\over 2} \over 2}
={F(z) \over 2}
=F \left({z\over 2}\right)$$
(by the convexity of $F$ applied to the midpoint ${z\over 2}$
between
${p+z\over 2}$ and ${q+z\over 2}$ together with the
homogeneity of $F$).  Since this is false, our assertion is
proved.
\medskip
{\bf Step 2.}  Suppose $p \in \partial \B^n(0, \, 1)$ with
$F(p) = G(p) = 0$.  Then there is a number $1 < \lambda <
\infty$ such that 
$$F(q) \le \lambda G(q) \le \lambda^2 F(q)$$
for each $q \in \R^n$.
Outside a small neighborhood of $p$ we infer from Step~1
and the continuity of $F$ and $G$ 
that the restrictions of both $F$ and $G$ to $\partial
\B^n(0, \, 1)$ are uniformly bounded away from zero and
infinity so that choice of such  a $\lambda$ is possible.
We also infer from Taylor's theorem and our assumptions about
continuity and  positivity of second derivatives that
the ratios of $F(q)$ and $G(q)$ are bounded above and below
for $q$'s which are near $p$ with $q\bullet p = 0$.  Our
assertion now follows from the assumed homogeneity of $F$ and $G$.
\medskip
{\bf Step 3.}  There is a number $1 < \lambda < \infty$ such that
$$\big( F - DF(p)\big)(q) \le \lambda
\big( G - DG(p)\big)(q) \le \lambda^2
\big( F - DF(p)\big)(q)$$ 
whenever $0 \not= p, \, q \in \R^n$
Indeed, for  each $p \not= 0$ the  functions
$\big( F - DF(p)\big)$ and $\big( G - DG(p)\big)$
satisfy the same general assumptions as do $F$ and $G$ but
also vanish at $p$.  For each fixed $p$ we infer the inequality of 
Step~3 from Step~2.  The arguments of Step~2 in fact give
uniform  estimates so that such an inequality holds
simultaneously for all $p \in \partial \B^n(0, \, 1)$.
The positive homogeneity assumed implies that such $p$'s are
sufficient for the general assertion.
\medskip
{\bf Step 4.}  Suppose $L \colon \R^n \to \R$ is linear.
Then
$$\int_X (L \circ h)\, d \varphi
=L\left(\int_X  h\, d \varphi\right)$$
for $X$ and $h$ as above.  To see this one approximates $h$ by
functions with finite images.
\medskip
{\bf Step 5.}  Suppose $X$ and $h$ are as above and
$p= \int_X h \, d \varphi$.  If $p = 0$ replace $p$ by any
other vector.  Set $F_\ast = F - DF(p)$ and
$G_\ast = G - DG(p)$ so that $F_\ast(p) = G_\ast(p) = 0$. 
We infer from Step~3 that the bounded ratios asserted in the
Proposition  hold if we replace $F$ by $F_\ast$ and $G$ by
$G_\ast$.  We infer from Step~4 that this is adequate to prove
the original bounded ratios asserted.  \E
\bigskip
{\bf C.5 Proposition (A criterion for vector functions to be nearly
constant).}
{\sl
Suppose
\smallskip\noindent
(a) $\varphi$ measures $X$ with $\varphi(X) < \infty$.
\smallskip\noindent
(b) $f \colon X \to \R^n$ is $\varphi$ summable so that
$W = \int_X f \, d \varphi \in \R^n$.
\smallskip\noindent
$W = \int_X f \, d \varphi \in \R^n$.
(c) $0 < \epsilon < 1$ and
$$
\int_X |f|\, d \varphi - \left| \int_Xf \, d \varphi\right|
\le \epsilon^2 \left|\int_X f \, d \varphi \right|.$$
(d) $Y = X \cap \big\{ x \colon f(x) \bullet W \le (1 - \epsilon)
|f(x)|\, |W|\big\}.$
\smallskip\noindent
Then
$$
\int_Y |f|\, d \varphi \le \epsilon \left| \int_X f \, d
\varphi\right|.$$
}
\bigskip
PROOF.
The case $W = 0$ is left to the reader.  For $W \not= 0$ we set
$u = W / |W|$ and, for $x \in X$, set
$$g(x) = f(x) \bullet u, \quad h(x) = \big(|f(x)|^2 -
g(x)^2\big)^{1 \over 2}.$$
Since
$$\left| \int_X f \, d \varphi\right| 
= u \bullet \int_X f \, d \varphi
= \int_X u \bullet f \, d \varphi
= \int_X g  \, d \varphi,$$
supposition (c) becomes
$$\int_X \big(g^2 + h^2\big)^{1 \over 2}\, d \varphi
\le \big(1 + \epsilon^2\big) \int_X g \, d \varphi$$
and our desired conclusion is
$$\int_Y \big(g^2 + h^2\big)^{1 \over 2}\, d \varphi
\le \epsilon \int_X g \, d \varphi.$$
We compute
$$\eqalign{\int_X g \, d \varphi
& 
= \int_Y g \, d \varphi
+ \int_{X \sim Y} g \, d \varphi
\cr &
\le (1 - \epsilon)
\int_Y \big(g^2 + h^2\big)^{1 \over 2} \, d \varphi
+ \int_{X \sim Y} \big(g^2 + h^2\big)^{1 \over 2} \, d \varphi
\cr &
\le \big(1 + \epsilon^2 \big)
 \int_X g \, d \varphi
- \epsilon  \int_Y \big(g^2 + h^2\big)^{1 \over 2} \, d \varphi\cr
}$$
so that
$$\epsilon \int_Y \big(g^2 + h^2\big)^{1 \over 2}\, d \varphi
\le \epsilon^2 \int_X g \, d \varphi$$
which implies our  conclusion. \E
\bigskip
{\bf C.6 Corollary (Another criterion for vector functions to be nearly
constant).}
{\sl
Suppose $F \colon \R^n \to \R^+$ is a convex function
which is twice continuously differentiable on $\R^n\sim \{0\}$.
Suppose also that $F$ is positively homogeneous of degree~1
and has positive second derivatives when
restricted to each straight line not containing the origin.
In case $n = 1$ we assume additionally that $F$ is not
everywhere equal to zero.
Then there is $\lambda > 0$ with the following property:
Suppose
\smallskip \noindent
(a) $\varphi$ measures $X$ with $\varphi(X) < \infty$.
\smallskip\noindent
(b) $f \colon X \to \R^n$ is $\varphi$ summable so that
$p = \int_X f \, d \varphi \in \R^n$.
\smallskip\noindent
(c) $0 < \epsilon < 1$ and
$$
\int_X F \circ f\, d \varphi - F\left( \int_Xf \, d \varphi\right)
\le \lambda \epsilon^2 \left|\int_X f \, d \varphi \right|.$$
(d) $Y = X \cap \big\{ x \colon f(x) \bullet p \le (1 - \epsilon)
|f(x)|\, |p|\big\}.$
\smallskip\noindent
Then
$$
\int_Y |f|\, d \varphi \le \epsilon \left| \int_X f \, d
\varphi\right|.$$
}
\bigskip
PROOF.
Set $G(v) = |v|$ in Proposition~C.4  and use Proposition~C.5. \E
\bigskip
{\bf PROOF OF THEOREM C.3}
We will treat only the case in which $\Psi$ does not depend on
its $\R^n$ variables and leave the general case to the
reader.  Initially we consider the case in  which $T$ and  the
$T_j$'s are general $k$ currents representable by integration
(not  necessarily  rectifiable).
Suppose then that we set $N= {n \choose k}$ and
identify $\wedge_k\R^n$ with $\R^N$.  We use our ellipticity
assumption to infer the existence of a convex function
$$F\colon \R^N \to \R^+\quad\hbox{with}\quad F | \G_0(n, \, k) = \Psi$$
which is positive and twice continuously differentiable
on $\R^N\sim \{0\}$ and which is positively homogeneous of degree~1
and has positive second derivatives when
restricted to each straight line not containing the origin.
By the integral of $F$ over 
$T$ we mean the number
$$\F(T) = \int_{{\bf R}^n} F\big(\vec T\big)\, d \|T\|.$$
we leave it to the reader to use the convexity of $F$ to verify that
$$\F(T) = \sup\bigg\{ T(\omega) \colon \omega(x) \le F
\hbox{ for each }x \bigg\}.$$
We infer from this estimate that
$$F(T) \le \lim\inf_j F(T_j).$$
Associated with $T$ and each smooth $\varphi \colon \R^n \to
\R^+$ having  compact support is the current
$T\rightangle \varphi$, given by setting
$$ (T\rightangle \varphi) (\omega) =
 T(\varphi \wedge \omega)$$
for each $\omega$.  Clearly
$$ T \rightangle \varphi = (\| T \| \rightangle \varphi) \wedge
\vec T.$$
Corresponding to each $\epsilon > 0$ we can apply Besicovitch's
covering theorem [FH 2.8.18] to balls
centered in the Lebesgue set of $\|T\|$ [FH 2.9.9] to infer the
existence of $\|T\|$ Lebesgue points
points $p_1, \, p_2, \, \ldots, \, p_M$ for $ \vec T$ and
radii $r_1, \, r_2, \,  \ldots, \, r_M$
and smooth functions
$\varphi_1,\, \varphi_2, \, \ldots, \, \varphi_M$
and
$$\varphi_0 = 1 - \sum_{j-1}^M \varphi_j.$$
having the  following properties:
\smallskip\noindent
(i) $|p_i - p_j| > r_i + r_j$ for each $i \not=j$;
\smallskip\noindent
(ii) each $\varphi_i$ is nearly the characteristic function of
$\B^n(p_i, \, r_i)$;
\smallskip\noindent
(iii) for each $i = 1, \, \ldots, M$,
$$\int_{x \in {\bf R}^n} \varphi_i(x) \big( \vec T(x) - \vec
T(p_i)\big) \, d \|T\|x < \epsilon \int_{{\bf R}^n} \varphi_i \,
d \|T\|.$$
$$\int_{{\bf R}^n}\varphi_0 \, d \|T\| < \epsilon;
\leqno{(iv)}$$
We then have
\smallskip\noindent
(1) For each $i = 0, \, \ldots, \, M$
the weak convergence of our currents implies
$$\int_{{\bf R}^n} \vec T_j \, d \| T_j  \rightangle \varphi_i \|
\to
\int_{{\bf R}^n} \vec T \, d \| T \rightangle \varphi_i \|
\quad\hbox{as } j \to \infty;$$
\smallskip\noindent
(2) If $\epsilon$ is very small then, for all sufficiently large $j$'s
and for each $i = 1, \, \ldots, \, M$,
$$\eqalign{\int_{{\bf R}^n} \vec T_j \, d \| T_j \rightangle \varphi_i\|
& \approx
\int_{{\bf R}^n} \vec T \, d \| T \rightangle \varphi_i\|
\cr &
\approx
\int_{{\bf R}^n} \vec T(p_i) \, d \| T \rightangle \varphi_i\|
\cr &
=
\vec T(p_i) \| T \rightangle \varphi_i\| \R^n.
\cr}$$
Now suppose additionally that $\F(T_j) \to \F(T)$ as $J \to
\infty$.  Then 
\smallskip\noindent
(3) For each $i = 0, \, \ldots, \, M$, the convergence of our
$\F$ integrals implies
$$\int_{{\bf R}^n} F(\vec T_j) \, d \| T_j  \rightangle \varphi_i \|
\to
\int_{{\bf R}^n} F(\vec T) \, d \| T \rightangle \varphi_i \|
\quad\hbox{as } j \to \infty;$$
(4) If $\epsilon$ is very small then, for
all sufficiently large $j$'s and for each $i = 1, \,
\ldots, \, M$,
$$\eqalign{\int_{{\bf R}^n} F(\vec T_j) \, d \| T_j \rightangle 
\varphi_i\|
&
\approx
\int_{{\bf R}^n} F(\vec T) \, d \| T \rightangle \varphi_i\|
\cr&
\approx
\int_{{\bf R}^n} F(\vec T(p_i)) \, d \| T \rightangle \varphi_i\|
\cr&
=
\vec T(p_i) \| T \rightangle \varphi_i\| \R^n.
\cr}$$
\smallskip\noindent
(5) If $\epsilon$ is very small we 
combine (2) and (4) above to conclude for large $j$'s
and each $i= 1, \, \ldots, \, M$,
$$\int_{{\bf R}^n} \vec T_j \, d \| T_j \rightangle \varphi_i\|
\approx F\left(
\int_{{\bf R}^n} F(\vec T_j) \, d \| T_j \rightangle \varphi_i\|
\right)$$
so that
$$\int_{{\bf R}^n} \vec T_j \, d \| T_j \rightangle \varphi_i\|
-F\left(
\int_{{\bf R}^n} F(\vec T_j) \, d \| T_j \rightangle \varphi_i\|
\right)$$
is very small.
\smallskip\noindent
(6) For large $j$ and each $i = 1, \, \ldots, \, M$ we set
$$W_{ij} =
\int_{{\bf R}^n} \vec T_j \, d \| T_j  \rightangle \varphi_i \|
, \quad w_{ij} = {W_{ij}\over \left| W_{ij}\right|}$$
and, in case $\epsilon$ is very small
infer from (5) Corollary~C.6 that, except for very small
measure, the $k$~vectors $\vec T_j(x)$, $x \in \B^n(p_i, \
r_i)$ are very close to~$w_{ij}$.  We then use (2) again to
infer that $\|T_j \rightangle \varphi_i\|\R^n$ is very close
to $\|T \rightangle \varphi_i\|\R^n$.
\smallskip\noindent
(7) In case $\epsilon$ is  very small, we
infer from (6) that for each $i = 1, \, \ldots, \, M$
the  varifolds
$${\bf v}\big(S_j, \, \theta_j \, \varphi_i, \, \sigma_j\big)
\quad\hbox{and}\quad
{\bf v}\big(S, \, \theta \, \varphi_i, \, \sigma\big)$$
are weakly close for large $j$.
Again, if $\epsilon$ is very small we 
infer from (3) and the fact that the restriction of $F$ to
$\G_0(n, \, k)$ is bounded above and below that the varifolds
$${\bf v}\big(S_j, \, \theta_j \, \varphi_i, \, \sigma_j\big)
\quad\hbox{and}\quad
{\bf v}\big(S, \, \theta \, \varphi_i, \, \sigma\big)$$
are weakly close to~$0$ for large~$j$.  Since our choice of
$\epsilon$ was arbitrary, we infer that
$${\bf v}\big(S_j,\, \theta_j, \, \sigma_j\big) \h
{\bf v}\big(S,\, \theta, \, \sigma_j\big)\quad\hbox{as } j \to
\infty.$$ 
This is the assertion of Theorem~C.3  \E
\bigskip
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\end
