paper - Universität Regensburg
96
Transcript of paper - Universität Regensburg
ON MATHEMATICAL MODELS FOR PHASE
SEPARATION IN ELASTICALLY STRESSED SOLIDS
HARALD GARCKE
Contents
1. Introduction 2
2. The di�use interface model 7
3. Existence for the di�use interface system 12
3.1. The gradient ow structure 12
3.2. Assumptions 15
3.3. Weak solutions 16
3.4. The implicit time discretisation 17
3.5. Uniform estimates 21
3.6. Proof of the existence theorem 25
3.7. Uniqueness for homogeneous linear elasticity 26
4. Logarithmic free energy 29
4.1. A regularised problem 32
4.2. Higher integrability for the strain tensor 36
4.3. Higher integrability for the logarithmic free energy 42
4.4. Proof of the existence theorem 45
5. The sharp interface limit 46
5.1. The �{limit of the elastic Ginzburg{Landau energies 52
5.2. Euler{Lagrange equation for the sharp interface functional 60
6. The Gibbs{Thomson equation as a singular limit in the
scalar case 70
7. Discussion 79
8. Appendix 81
9. Notation 86
References 90
1
1. Introduction
We study a mathematicalmodel describing phase separation in multi{
component alloys in the presence of elastic interactions. Our aim is to
describe the following phenomena. At temperatures larger than a crit-
ical temperature �
c
, one homogenous phase characterised by a certain
mixture of the alloy components is energetically favourable. If the
temperature is decreased below �
c
two or more phases develop and the
system rapidly decomposes into a microstructure consisting of many re-
gions with di�erent phases (see Figure 1). Each phase is characterised
by a distinct composition of the individual components and these are
such that the bulk chemical energy becomes small. The formation of
the microstructure is on a very fast time scale and is called spinodal
decomposition or phase separation. One observes the development of
many connected regions of distinct phases which we call particles.
Associated with the interfaces between phases is an interfacial en-
ergy. Since the phases are �nely mixed, the interfacial energy is large
after spinodal decomposition. In a second stage of the evolution, the
microstructure coarsens and hence the interfacial energy decreases. In
the case of negligible elastic e�ects, particles tend to become round,
small particles shrink and larger ones grow (see Figure 2). This part of
the evolution called coarsening (or Ostwald ripening) is mainly driven
by the reduction of the total amount of interfacial area.
If the components of the mixture have di�erent elastic moduli or
di�erent lattice structure, elastic e�ects might in uence the rate of
coarsening and the morphology of the particles. Elastic e�ects can
result for example from di�erent lattice spacings of the alloy compo-
nents. In Figure 3 we demonstrate the e�ects that anisotropic elastic
energy and di�erent lattice spacings can have on the coarsening mor-
phology. The elastic e�ects become more important at later stages of
the evolution. This can be seen by comparing the energy of the elastic
and surface energy (see Fratzl, Penrose, Lebowitz [40]). Furthermore
numerical simulations indicate this.
In applications of multi{component alloys it is important to know
the rate of coarsening. The famous theory of Lifshitz and Slyozov [68]
and Wagner [96] (LSW{theory) says that the average particle size in
systems driven by di�usion without stress e�ects is proportional to
t
1
3
. If stress e�ects cannot be neglected then approaches such as the
LSW{theory does not seem to be possible and the late stage evolution
2
Figure 1. Spinodal decomposition in a binary system.
A perturbation around a homogeneous state decomposes
into a con�guration with regions of di�erent phases. The
concentration of one of the components is shown.
in phase separating systems with elastic e�ects is not yet very well
understood.
Material scientists hoped to �nd situations in which stress e�ects can
slow down the rate of coarsening. Some experiments even suggested
that inverse coarsening is possible, i.e. the average domain size de-
creases (see e.g. [39, 40, 57, 61, 76, 75, 82, 83]). The understanding of
the mechanisms responsible for these e�ects is very important for tech-
nological applications. A uniform distribution of regions of di�erent
phases is desirable in order to guarantee evenly distributed material
properties of the sample. For example mechanical properties such as
the strength and the mechanical stability of the material depend on
3
Figure 2. Phase separation and coarsening in the
Cahn{Hilliard model without elasticity. The times from
left to right top to bottom are: t=0.02, 0.05, 0.2, 0.5,
1.0, 2.0, 4.0, 8.0.
Figure 3. Phase separation and coarsening in the
Cahn{Hilliard model in the case of homogeneous elas-
ticity with cubic symmetry. The times from left to right
top to bottom are: t=0.02, 0.05, 0.2, 0.5, 1.0, 2.0, 4.0,
8.0.
how �nely mixed regions of di�erent phases are. The control of ag-
ing and therefore of the lifetime of materials depends on the ability
4
to understand the coarsening process. This demonstrates the impor-
tance of a reliable mathematical model to describe phase separation in
multi{component solids in the presence of elastic interactions.
In Section 2 we introduce a model for phase transformations in solids
taking elastic e�ects into account. The free energy of the system is a
Ginzburg{Landau functional de�ned in terms of the concentrations of
the individual components and the displacements of the lattice. Inter-
faces are described by a transition layer of �nite but small thickness.
Two terms in the Ginzburg{Landau free energy are classical. The �rst
one is quadratic in the gradients of the concentrations and penalises
large gradients. The other represents the bulk chemical energy and in
the case of phase separation its density is a non-convex function of the
concentrations. In order to model e�ects due to stresses we also include
elastic terms in the Ginzburg{Landau free energy. Since in the phe-
nomena we are interested in, the strain is typically small, we present a
geometrically linear theory formulated in terms of the linearised strain
tensor.
The evolution of the system is governed by di�usion equations for the
concentrations of the individual components and a quasi-static equilib-
rium for the mechanical part. The latter condition is reasonable since
the mechanical equilibrium is obtained on a much faster time scale
than di�usion takes place. The overall system consists of a system of
fourth order parabolic di�usion equations coupled to an elliptic system
describing mechanical equilibrium. In the two component case with no
elastic interactions the elliptic{parabolic system reduces to the stan-
dard scalar Cahn{Hilliard equation.
In the third section we prove an existence result for the elliptic{
parabolic system in the case that the terms de�ning the free energy are
smooth. In order to obtain approximate solutions of the problemwe use
the method of implicit time discretisation and at each time step solve
a minimum problem taking into account the variational structure. A
major di�culty of the analysis is the fact that the displacement gradient
enters the di�usion equation quadratically. Hence, it is necessary to
show strong convergence of the displacement gradient in L
2
. We also
state a uniqueness result in the case case of linear elasticity under
the assumption of homogeneous elasticity, i.e. the elasticity tensor is
independent of the concentrations.
Free energies derived from a mean �eld theory contain logarithmic
entropy terms. As a result the elliptic{parabolic system contains sin-
gular terms. Section 4 is devoted to the mathematical analysis of the
singular elliptic{parabolic system. To prove existence of solutions a
higher integrability result for the gradient of solutions to the elastic
5
system is needed. We prove an L
p
{estimate for the gradient for some
p > 2 using methods introduced by Giaquinta and Modica [46] (see also
[45]). These are based on Caccioppoli inequalities and reverse H�older
inequalities. Having derived higher integrability of the displacement
gradient enables us to obtain a priori estimates for the singular terms
in the di�usion equation which is crucial for the existence proof.
In the �fth section we study the small interfacial thickness limit of the
Ginzburg{Landau free energy. As mentioned above, in the Ginzburg{
Landau theory the interface is modelled by an interfacial layer with a
positive but small thickness. In sharp interface models the interface
is described by hypersurfaces on which certain quantities su�er jump
discontinuities. In the Ginzburg{Landau free energy we introduce a
scaling parameter " which is proportional to the interfacial thickness.
As " tends to zero, we show that in the sense of �{limits the func-
tionals converge to a certain functional, de�ned for vector functions of
bounded variation. This generalises results of Modica [77], Baldo [8]
and Ambrosio [7] to the case with elastic contributions. We also derive
the Euler{Lagrange equation for the sharp interface energy functional.
These equations include mechanical equilibrium conditions in the bulk
and across the interfaces, a modi�ed Gibbs{Thompson law at inter-
faces and Young's law at junctions of interfaces and at points where an
interface intersects the outer boundary.
In Section 6 we analyse the limiting behaviour of the Euler{Lagrange
equation for minimisers of the elastic Ginzburg{Landau energies as the
interfacial thickness tends to zero. We consider the binary case, i.e.
only two components are present. It is shown that the Lagrange multi-
plier entering the Euler{Lagrange equation converges and in the sharp
interface limit we obtain a weak formulation of the Gibbs{Thomson
relation. The �rst variation of the sharp interface functional includes
an elastic term (the so called Eshelby tensor) which is quadratic in the
strains and has to be evaluated at the free boundary. This involves
traces of the strain tensor. Since in general not enough regularity is
known to obtain the existence of these traces, we give a weak formula-
tion involving only bulk contributions of the elastic terms.
In Section 7 we brie y summarise our results and state some open
questions. For the convenience of the reader we collected in Section
8 results that are used in the text. A reader not familiar with our
notation is referred to Section 9 where we list our notation.
All numerical computations presented in this thesis have been per-
formed with a numerical method that has been developed in coopera-
tion with Martin Rumpf and Ulrich Weikard [42] (see also [99]). The
6
implementation of the algorithm is due to Ulrich Weikard who also
performed all computations.
It is a pleasure to thank my academic teacher Hans Wilhelm Alt
for his support over the years and for all which I was able to learn
from him. I am grateful to Luigi Ambrosio, Nicola Fusco and Barbara
Niethammer for helpful discussions. Luise Blank, Barbara Niethammer
and Vanessa Styles read a preliminary version of this thesis and many
constructive comments helped to improve the presentation. The work
with Martin Rumpf and UlrichWeikard on numerical aspects of Cahn{
Hilliard systems with elasticity helped to understand the problem and
my thanks go to them.
2. The diffuse interface model
Assume the alloy consists of N components. We denote by c
k
(k =
1; :::; N) the concentration of component k and therefore, the vector
c = (c
k
)
k=1;:::;N
has to ful�l the constraint
P
N
k=1
c
k
= 1, i.e. c lies in
the a�ne hyperplane
� := fc
0
= (c
0
k
)
k=1;:::N
2 IR
N
j
N
X
k=1
c
0
k
= 1 g:
To describe elastic e�ects we de�ne the displacement �eld u(x), i.e. a
material point x in the undeformed body will be at the point x+u(x)
after deformation. Since in phase separation processes the displacement
gradient usually is small, we consider an approximative theory based
on the linearised strain tensor
E(u) =
1
2
�
ru+ (ru)
t
�
:
A generalised Ginzburg{Landau free energy taking elastic e�ects into
account is of the form
E(c;u) =
Z
�
1
2
rc : �rc+(c) +W (c; E) +W
�
(E)
�
(1)
where � IR
n
, n 2 IN, is a bounded domain with Lipschitz boundary.
The �rst term in the energy is the gradient part penalising rapid spatial
variations in the concentrations. The tensor � mapping IR
N�n
into
itself is assumed to be symmetric and positive de�nite. The second
7
summand is the homogeneous (or \coarse grain") free energy density
at zero stress taking into account the chemical energy of the system.
The function depends on temperature and is convex above a critical
temperature �
c
and non{convex for temperatures less than the critical
temperature. In the latter case has several local minimisers giving
rise to the appearance of di�erent phases. These �rst summands in the
total free energy E are classical contributions to a Ginzburg{Landau
free energy. Energies consisting of two terms of this form go back to
van der Waals [95]. In the theory of phase separation in alloys they
have been introduced by Cahn and Hilliard [19].
The last two terms in the free energy take elastic e�ects into account.
The term W (c; E) is the elastic free energy density and a typical form
is
W (c; E) =
1
2
(E � E
?
(c)) : C(c) (E � E
?
(c)) : (2)
Here, C(c) is the concentration dependent elasticity tensor mapping
symmetric tensors in IR
n�n
into itself. We require C(c) to be symmetric
and positive de�nite. The quantity E
?
(c) is the symmetric stress free
strain (or eigenstrain) at concentration c. This is the value the strain
tensor attains if the material were uniform with concentration c and
unstressed. If the vector c is equal to one of the standard cartesian
basis vectors e
1
; :::; e
N
then the system is equal to a pure component.
In this case E
?
(e
k
) is the value of the strain tensor if the material
consists only of component k and is unstressed. The function E
?
is a
suitable extension to all of �. Usually a linear extension of the form
E
?
(c) =
N
X
k=1
c
k
E
?
(e
k
)
is assumed (Vegard's law). The elastic energy density (2) is the stan-
dard choice which goes back to the early work of Eshelby [33] and
Khachaturyan [58] (see also [65, 40]). However, we will obtain results
for more general densities.
The remaining term W
?
(E) represents energy e�ects due to exter-
nally applied forces. For simplicity, we assume that there are no body
forces and that the boundary tractions are dead loads given by a con-
stant and symmetric tensor S
�
, i.e. the tractions applied to @ are
given by S
�
n. The work needed to bring the undeformed body into the
state with displacement u : ! IR
3
is then given by
�
Z
@
u � S
�
n = �
Z
ru : S
�
= �
Z
E(u) : S
�
;
8
where n is the outer unit normal to . Hence,
W
�
(E
0
) := �E
0
: S
�
is the energy density of the applied outer forces.
To describe evolution phenomena in the system, we consider mass
di�usion for the individual components leading to di�usion equations
for the concentrations. Mechanical equilibrium is attained on a much
faster time scale than di�usion takes place. Therefore, we will assume
a quasi{static equilibrium for u, i.e. for all times
r � S = 0;
where
S = W
;E
(c; E(u)):
We remark that the solution of the elastic system in general depends
on time since c in general is time dependent. It is always assumed
that W depends on its second argument only through its symmetric
part, i.e. W (c
0
; E
0
) = W (c
0
; (E
0
)
t
). This implies that S = W
;E
(c
0
; E
0
) is
symmetric.
The di�usion equations for the concentrations c
k
(k = 1; :::; N) are
based on mass balances for the individual components. To de�ne the
mass balance we need to introduce chemical potentials �
k
which are
de�ned as the variational derivative of the total free energy E with
respect to c
k
, i.e.
�
k
= �(r � �rc)
k
+
;c
k
(c) +W
;c
k
(c;u):
Now Onsager's postulate [85, 86, 60] says that each thermodynamic ux
is linearly related to every thermodynamic force. Since in our case the
thermodynamic forces are the negative chemical potential gradients,
we obtain the phenomenological equations (see Kirkaldy and Young
[60], p. 137)
J
k
= �
N
X
l=1
L
kl
r�
l
(3)
with a constant matrix L = (L
kl
)
k=1;:::;N ;l=1;:::;N
2 IR
N�N
. The Onsager
reciprocity law (see [60], p. 137, and [85, 86]) states that the matrix L
has to be symmetric, which we assume in the following. Having de�ned
the ux, the di�usion equations follow from the balance of mass as
@
t
c
k
= �r � J
k
: (4)
To ensure that the di�usion equations (4) are consistent with the con-
straint
P
N
k=1
c
k
= 1 the uxes have to ful�l a linear dependency of the
9
form
N
X
k=1
J
k
= 0: (5)
Since the identities (3) and (5) have to hold for all possible chemical
potentials we have to impose
N
X
l=1
L
kl
= 0: (6)
This property of the mobilitymatrixL = (L
kl
)
k=1;:::;N ;l=1;:::;N
we assume
from now on. As a consequence the di�usion equations (4) become
@
t
c
k
= r �
N
X
l=1
L
kl
r�
l
!
=
N
X
l=1
L
kl
�
1
N
N
X
m=1
(�
l
� �
m
) :
Hence, the di�usion equations can be expressed via the chemical poten-
tial di�erences (�
l
� �
k
). In particular, the evolution can be described
via the vector of generalised chemical potential di�erences
w =
1
N
N
X
m=1
(�
l
� �
m
)
!
l=1;:::;N
= P�
where P is the euclidian projection of IR
N
onto
T� = fd
0
= (d
0
k
)
k=1;:::N
2 IR
N
j
N
X
k=1
d
0
k
= 0 g
which is the tangent space to �. A simple computation yields that
w is the variational derivative of E when one takes the constraint
P
N
k=1
c
k
= 1 into account. In fact, introducing e = (1; :::; 1), we get
w = � �
1
N
(� � e)e where the second term is the Lagrange multiplier
associated to the constraint
P
N
k=1
c
k
= 1.
Altogether we obtain the system of equations
@
t
c = L�w; (7)
w = P (�r � �rc+
;c
(c) +W
;c
(c; E(u))) ; (8)
r � S = 0; (9)
S = W
;E
(c; E(u)): (10)
10
If only two components are present, one can use the constraint c
1
+
c
2
= 1 to reduce the system for the concentrations to a single equation.
In this case the equations (7){(10) were �rst stated by Larch�e and Cahn
[63] for = 0 and for nonzero by Onuki [87].
As boundary conditions we impose no{ ux conditions for the J
k
and
the natural boundary conditions which one obtains from variations of
the energy functional with respect to c and u. Therefore,
Lrw � n = 0; (11)
�rc � n = 0; (12)
S � n = S
�
� n: (13)
In addition, we impose initial conditions for c, i.e.
c(x; 0) = c
0
(x) (14)
for a given function c
0
with c
0
(x) 2 � for all x 2 . Since we assume
that the mechanical equilibrium is obtained instantaneously no initial
conditions for u are needed. We remark that the no{ ux boundary
condition implies that the total mass of the solution to (7){(14) is a
conserved quantity, i.e. for all t > 0
Z
c(x; t)dx =
Z
c
0
(x)dx:
Other boundary conditions are possible. For example we could im-
pose Dirichlet conditions for u on parts of the boundary @. This
means to prescribe the deformation on parts of the boundary and hence
a unique u could be determined. The boundary condition (13) on the
other hand prescribes u only up to in�nitesimal rigid displacements
(i.e. translations and in�nitesimal rotations). This is typical for prob-
lems in elasticity that are based on a linearised strain tensor E. The
non{uniqueness in u will have no e�ect on the evolution of c since only
E(u) enters the equation for w.
The above system of equations is thermodynamically consistent if
the second law of thermodynamics holds. For isothermal systems the
second law is equivalent to a dissipation inequality for the free energy.
Using the equations (7){(10) and the symmetry of S and S
�
we com-
pute formally:
@
t
�
1
2
rc : �rc+(c) +W (c; E) +W
�
(E)
�
=
= �rc : r@
t
c+
;c
� @
t
c+W
;c
� @
t
c+W
;E
: @
t
E � S
�
: @
t
E
= r � f�rc � @
t
c+ S@
t
u �S
�
@
t
ug+ f�r � �rc+
;c
+W
;c
g � @
t
c
= r � f�rc � @
t
c+ S@
t
u �S
�
@
t
u� J � �g �r� : Lr�:
11
Hence, assuming that L is positive semide�nite we obtain the local
dissipation inequality
@
t
�
1
2
rc : �rc +(c) +W (E; c) +W
�
(E)
�
� (15)
� r � f�rc � @
t
c+ S@
t
u� S
�
@
t
u� J � �g :
This shows that the rate of change of energy cannot exceed the total
power expended plus the energy in ow. In inequality (15) the term
S@
t
u � S
�
@
t
u denotes the power expended by deformation stresses,
J � � accounts for the energy in ow by di�usion and �rc � @
t
c can be
interpreted to result from ow of energy due to moving phase inter-
faces (see [6, 49, 50]). Integrating the dissipation inequality (15) and
using the boundary conditions shows that the free energy serves as a
Lyapunov functional. In the following section the Lyapunov property
of the free energy will give the main a priori estimate in the existence
theory.
3. Existence for the diffuse interface system
In this section we prove an existence result for the elliptic{parabolic
system. We use the fact that the system aims to decrease the free
energy. In the concentration variables the system has a gradient ow
structure with respect to a weighted
�
H
1
(; IR
N
)
�
�
{scalar product and
for the mechanical part the relaxation is in�nitely fast and equilibrium
is achieved instantaneously. Using this property of the system we make
an implicit time discretisation which leads to a variational problem
in each time step. To formulate the variational problem we need to
understand in which sense the system can be understood as the steepest
descent of the free energy. Therefore, we introduce a suitably weighted
scalar product in the dual of H
1
.
3.1. The gradient ow structure. Let
X
1
:= fc 2 H
1
(; IR
N
) j c 2 � almost everywhere g
be the space of all H
1
{functions ful�lling the constraint for the con-
centrations. Since the system (7){(13) preserves the mean value of the
vector c, it will also be useful to introduce the spaces
X
m
1
:= fc 2 X
1
j �
Z
c
k
= m
k
g
12
where m = (m
k
)
k=1;:::;N
is a constant vector of mean values with
1
N
P
N
k=1
m
k
= 1. We de�ne the tangent space
Y = fz 2 H
1
(; IR
N
) j
Z
z = 0;
N
X
k=1
z
k
= 0 g
of X
m
1
and the space of linear functionals D on H
1
(; IR
N
) that vanish
on the L
2
{complement of Y
D = ff 2
�
H
1
(; IR
N
)
�
�
j hd; fi
H
1
;(H
1
)
�= 0 for all d = d(x)(1; :::; 1)
where d is a scalar valued function with d 2 H
1
()
and for all d � e
k
; k = 1; :::; N g:
Then we introduce the mapping L associated to the di�erential oper-
ator z 7! �L�z as a mapping from Y to D via
L(z)(�) =
Z
Lrz : r�:
The fact that L(z) 2 D follows from (6) and the de�nition of Y . Later
we need the inverse of the mapping L which we denote by G. The
invertibility of L follows from the Poincar�e inequality and the Lax{
Milgram theorem provided that L is positive de�nite on T�. It holds
(LrGf ;r�)
L
2
= h�; fi
for all � 2 H
1
(; IR
N
) and f 2 D.
For all f
1
; f
2
2 D we de�ne the scalar product
(f
1
; f
2
)
L
:= (LrGf
1
;rGf
2
)
L
2
and the corresponding norm
kfk
L
=
p
(f ; f)
L
for all f 2 D:
Since all functions
�
f 2 Y can be interpreted as elements in D via the
mapping � 7!
R
�
f � � for � 2 H
1
(; IR
N
) the scalar product (:; :)
L
and the norm k:k
L
are also de�ned for functions in Y . Using Young's
inequality we obtain that for all � > 0 and all d 2 Y
kdk
2
L
2
= (LrGd;rd)
L
2
� kL
1
2
rGdk
L
2kL
1
2
rdk
L
2(16)
�
C
L
�
kdk
2
L
+ �krdk
2
L
2
;
where C
L
is a constant depending on L.
Introducing the Lagrange multiplier�(t) =
�
�
R
w
�
(t) we can rewrite
the equation (7) as
G@
t
c = � �w:
13
Then the equations (7), (8) lead to
(�rc;r�)
L
2
+ (
;c
(c); �)
L
2
+ (W
;c
(c; E(u)); �)
L
2
=
= (w; �)
L
2
= � (G@
t
c; �)
L
2
= � (LrG@
t
c;rG�)
L
2
= � (@
t
c; �)
L
which holds for all � 2 Y . The left hand side in the above computation
is the di�erential D
c
E(c;u) in the direction �. Consequently,
h�;D
c
E(c;u)i = � (�; @
t
c)
L
for all � 2 Y; (17)
which means that (7), (8) can be interpreted as the steepest descent of
E in X
m
1
with respect to the (:; :)
L
scalar product and the variable c.
The fact that the evolution for c has this gradient ow structure will
be used in the implicit time discretisation.
Since we consider boundary conditions of the second type, it will
turn out that for �xed concentrations c the displacement u is only de-
termined up to translations and in�nitesimal rotations, i.e. up to in�n-
itesimal rigid displacements. We note that the strain E is uniquely de-
termined and hence in the evolution equation for c the non-uniqueness
of u will not play any role. This discussion motivates the introduction
of the space
X
2
:= fu 2 H
1
(; IR
n
) j (u;v)
H
1= 0 for all v 2 X
ird
g = X
?
ird
;
where
X
ird
:= fu 2 H
1
(; IR
n
) j there exist b 2 IR
n
and a skew symmetric
A 2 IR
n�n
such that u(x) = b+Axg;
is the space of all in�nitesimal rigid displacements. In X
2
we �x the
freedom of u with respect to in�nitesimal rigid displacements. With
this notation a weak formulation of (9) and (10) can be written as
h�;D
u
E(c;u)i = 0 for all � 2 X
2
; (18)
where D
u
E is the derivative of E with respect to u in the space X
2
.
Since W
;E
(c; E) is symmetric if E is symmetric we can also allow for
arbitrary � 2 H
1
(; IR
n
) in (18).
We remark that the above discussion was of formal nature but can
be justi�ed rigorously under suitable conditions on , W , c and u.
14
3.2. Assumptions. We state the following assumptions:
(A1) � IR
n
is a bounded domain with Lipschitz boundary,
(A2) the gradient energy tensor � is a constant, symmetric, positive
de�nite linear mapping of IR
N�n
into itself, i.e. in particular there
exists a constant
0
> 0 such that
A
0
: �A
0
�
0
jA
0
j
2
;
for all A
0
2 IR
N�n
,
(A3) the homogeneous free energy density can be written as
(c
0
) =
1
(c
0
) +
2
(c
0
) for all c
0
2 IR
N
with
1
;
2
2 C
1
(IR
N
; IR) and
1
convex. In addition, we assume
(A3.1)
1
� 0,
(A3.2) for all � > 0 there exists a C
�
> 0 such that
j
1
;c
(c
0
)j � �
1
(c
0
) + C
�
for all c
0
2 �;
(A3.3) there exists a constant C
1
> 0 such that
j
2
;c
(c
0
)j � C
1
(jc
0
j+ 1) for all c
0
2 �;
(A4) for the elastic energy densityW 2 C
1
(IR
N
� IR
n�n
; IR) we assume
(A4.1) W (c
0
; E
0
) only depends on the symmetric part of E
0
2 IR
n�n
,
i.e. W (c
0
; E
0
) = W (c
0
; (E
0
)
t
) for all c
0
2 IR
N
and E
0
2 IR
n�n
,
(A4.2) W
;E
(c
0
; :) is strongly monotone uniformly in c
0
, i.e.
there exists a c
1
> 0 such that for all symmetric E
0
1
; E
0
2
2 IR
n�n
(W
;E
(c
0
; E
0
2
)�W
;E
(c
0
; E
0
1
)) : (E
0
2
� E
0
1
) � c
1
jE
0
2
� E
0
1
j
2
;
(A4.3) there exists a constant C
2
> 0 such that for all c
0
2 � and
all symmetric E
0
2 IR
n�n
jW (c
0
; E
0
)j � C
2
(jE
0
j
2
+ jc
0
j
2
+ 1),
jW
;c
(c
0
; E
0
)j � C
2
(jE
0
j
2
+ jc
0
j
2
+ 1),
jW
;E
(c
0
; E
0
)j � C
2
(jE
0
j+ jc
0
j+ 1),
(A5) the energy density of the applied forces is assumed to be of the
form W
�
(E
0
) = �E
0
: S
�
with a constant symmetric tensor S
�
,
(A6) the mobility matrix L = (L
kl
)
k=1;:::;N ;l=1;:::;N
is assumed to be
(A6.1) symmetric,
(A6.2) to ful�l
P
N
l=1
L
kl
= 0,
(A6.3) to be positive de�nite on T�,
(A7) the initial data c
0
2 X
1
are assumed to ful�l
R
(c
0
) <1.
Let us comment on the stated assumptions. The assumptions on �
and L guarantee that the system (7), (8) de�nes a semi-linear para-
bolic system of fourth order in the variable c. Assumption (A3) uses a
15
splitting of the homogeneous free energy density . A convex part
1
is allowed to have large growth for c
0
large. We only assume that the
derivative
1
;c
can be controlled by
1
itself. Any polynomial growth
for
1
is allowed. This includes the cases that appear in applications
besides the one which is considered in Section 4. Note that for example
an exponential growth in is not allowed and we remark that a growth
larger than a polynomial growth could be treated with a method anal-
ogous to the one we present in the following section. The non{convex
part
2
is allowed to grow such that
2
;c
is sub-linear. The assump-
tion (A4.1) guarantees that W
;E
(c; E
0
) is symmetric for all symmetric
E
0
2 IR
n�n
. The monotonicity (A4.2) ensures that the elastic part of
the equation de�nes a quasi{linear elliptic system in E. Since
W (c
0
; E
0
) = W (c
0
;0) +
Z
1
0
W
;E
(c
0
; tE
0
) : E
0
dt
we can use assumptions (A4.2) and (A4.3) to conclude that there exist
positive constants c
3
and C
3
such that
W (c
0
; E
0
) � c
3
jE
0
j
2
� C
3
�
jc
0
j
2
+ 1
�
for all c
0
2 � and all symmetric E
0
2 IR
n�n
.
In the existence proof we will formulate a variational problem whose
Euler{Lagrange equations are an implicit time discretisation of the
elliptic{parabolic system. The growth conditions on and W ensure
existence of a solution to the variational problem and enables us to
compute the corresponding Euler{Lagrange equations. We also note
that it would be enough to state assumptions on P
1
;c
, P
2
;c
and PW
;c
rather than
1
;c
,
2
;c
and W
;c
. The evolution law is stated solely on �
and hence andW can be chosen arbitrary in the direction orthogonal
to �.
3.3. Weak solutions. The main goal of Section 3 is to show existence
of a weak solution to the problem (7){(14). We will use the following
solution concept.
De�nition 3.1. (Weak solution) A triple
(c;w;u) 2 L
2
(0; T ;H
1
(; IR
N
)) � L
2
(0; T ;H
1
(; IR
N
)) � L
2
(0; T ;X
2
)
with P
;c
(c) 2 L
1
(
T
) is called a weak solution of (7){(14) if and only
if
(i)
�
Z
T
@
t
� � (c� c
0
) +
Z
T
Lrw : r� = 0 (19)
for all � 2 L
2
(0; T ;H
1
(; IR
N
)) with @
t
� 2 L
2
(
T
) and �(T ) = 0,
16
(ii)
Z
T
w � � =
Z
T
fP�rc : r� +P
;c
(c) � � +PW
;c
(c; E(u)) � �g
(20)
for all � 2 L
2
(0; T ;H
1
(; IR
N
)) \ L
1
(
T
; IR
N
), and
(iii)
Z
T
W
;E
(c; E(u)) : r� =
Z
T
S
�
: r� (21)
for all � 2 L
2
(0; T ;H
1
(; IR
n
)).
Now let us state the existence theorem that will be proved in the
following subsections.
Theorem 3.1. Assume (A1){(A7). Then there exists a weak solution
in the sense of De�nition 3.1 which has the following properties:
(i) c 2 C
0;
1
4
([0; T ];L
2
());
(ii) @
t
c 2 L
2
�
0; T ; (H
1
())
�
�
,
(iii) u 2 L
1
(0; T ;H
1
(; IR
n
)) :
Remark 3.1. Using the perturbation arguments of Section 4.2 it is
possible to show higher integrability of ru, i.e. there exists a p > 2
such that ru 2 L
1
(0; T ;L
p
(; IR
n
)). In two space dimensions this and
the Sobolev embedding then implies that u 2 L
1
�
0; T ;C
0;�
(; IR
2
)
�
for
some � > 0.
3.4. The implicit time discretisation. Let T > 0 be an arbitrary
but �xed time,M 2 IN and �t =
T
M
. Then the implicit time discreti-
sation of the system (7){(10) is given by
c
m
� c
m�1
�t
= L�w
m
; (22)
w
m
= P (�r � �rc
m
+
;c
(c
m
) +W
;c
(c
m
; E
m
)) ; (23)
r � S
m
= 0; (24)
S
m
= W
;E
(c
m
; E
m
); (25)
for (c
m
;w
m
;u
m
). By an upper indexm 2 f0; :::;Mgwe denote the time
discrete solution at time m�t and E
m
is an abbreviation for E(u
m
).
Assuming that c
0
2 X
1
is given we want to determine (c
m
;w
m
;u
m
)
and hence S
m
inductively by solving (22){(25) together with the bound-
ary conditions (11){(13) for (c
m
;w
m
;u
m
).
17
Analogously to the discussion leading to (17) one can derive
h�;D
c
E(c
m
;u
m
)i = �
�
�;
c
m
� c
m�1
�t
�
L
for all � 2 Y: (26)
Also we have
h�;D
u
E(c
m
;u
m
)i = 0 for all � 2 X
2
(27)
and we observe that (26) and (27) are the Euler{Lagrange equations
of the functional
E
m;�t
(d;v) := E(d;v) +
1
2�t
kd� c
m�1
k
2
L
: (28)
Our goal now is to show the existence of an absolute minimiser of E
m;�t
in the class X
m
1
�X
2
where m := �
R
c
0
.
Lemma 3.1. Assume (A1){(A6) and suppose c
m�1
2 X
m
1
.
Then there exists a minimiser of E
m;�t
in X
m
1
� X
2
provided that
�t 2
�
0;
0
8(C
1
+C
3
)
2
C
L
�
.
Proof. The existence of a minimiser can be shown by the direct
method. We outline the main steps. Using the assumptions (A2){
(A6) we deduce that there exists a constant C > 0 such that for all
(d;v) 2 X
m
1
�X
2
it holds
E
m;�t
(d;v) �
0
2
krdk
2
L
2
+
c
3
2
kE(v)k
2
L
2
+
1
2�t
kd� c
m�1
k
2
L
� (C
3
+ C
1
)kdk
2
L
2
� C:
Since d� c
m�1
2 Y we can use (16) and the fact that c
m�1
is bounded
in H
1
(; IR
N
) to conclude
E
m;�t
(d;v) �
�
0
2
� (C
1
+ C
3
)�
�
krdk
2
L
2
+
+
c
2
2
kE(v)k
2
L
2
+
�
1
2�t
�
(C
1
+ C
3
)C
L
�
�
kd� c
m�1
k
2
L
� C
where C depends on c
m�1
. Since we assumed �t <
0
8(C
1
+C
3
)
2
C
L
we
can choose � =
0
4(C
1
+C
3
)
and obtain with the help of inequalities of
Korn (Theorem 8.3) and Poincar�e, that E
m;�t
is coercive on the space
X
m
1
�X
2
.
The space X
m
1
�X
2
is non{empty and hence we can choose a min-
imising sequence (d
�
;v
�
)
�2IN
with
1 > E
m;�t
(d
�
;v
�
)! inf
(d;v)2X
m
1
�X
2
E
m;�t
(d;v):
18
The coercivity of E
m;�t
implies that (d
�
;v
�
)
�2IN
is uniformly bounded
inX
m
1
�X
2
. Without loss of generality we can assume that (d
�
;v
�
)
�2IN
converges weakly inX
m
1
�X
2
to a limit (d;v) (otherwise we replace the
original sequence by a converging subsequence). Furthermore, we can
assume that d
�
converges strongly in L
2
(; IR
N
) and almost everywhere
in .
All terms in the energy besides
R
2
(d) and
R
W (d; E(u)) are con-
vex and therefore sequentially weakly lower semi-continuous on X
m
1
�
X
2
. Let us now discuss the remaining two terms. Condition (A3.3) im-
plies that
2
(c
0
) has at most a quadratic growth for large c
0
2 � and
hence the generalised dominated convergence theorem (see e.g. [2, 101])
of Lebesgue yields:
R
2
(d
�
)!
R
2
(d). It remains to show
Z
W (d; E(v)) � lim inf
�!1
Z
W (d
�
; E(v
�
)): (29)
Using the convexity of W (c
0
; :) (which follows from (A4.2)) we have
Z
fW (d
�
; E(v
�
))�W (d; E(v))g =
=
Z
fW (d
�
; E(v
�
))�W (d
�
; E(v))g+
Z
fW (d
�
; E(v))�W (d; E(v))g
�
Z
fW
;E
(d
�
; E(v)) : (E(v
�
) � E(v))g+
Z
fW (d
�
; E(v))�W (d; E(v))g :
The weak convergence of E(v
�
) in L
2
, the strong convergence of d
�
in
L
2
, the convergence almost everywhere of d
�
and the growth conditions
in (A4.3) then give (29). This yields
E
m;�t
(d;v) � lim inf
�!1
E
m;�t
(d
�
;v
�
)
and implies that (d;v) minimises E
m;�t
in X
m
1
�X
2
. ut
We choose the value at the new time step (c
m
;u
m
) as a minimiser
of E
m;�t
. In the following lemma we compute the Euler{Lagrange
equation associated with E
m;�t
which hold for (c
m
;u
m
).
Lemma 3.2. (Euler{Lagrange equations) The minimiser (c
m
;u
m
)
ful�ls
(i)
Z
�
c
m
� c
m�1
�t
�
� � +
Z
Lrw
m
: r� = 0 (30)
for all � 2 H
1
(; IR
N
).
19
(ii)
Z
w
m
� � =
Z
fP�ru
m
: r� +P
;c
(c
m
) � � +PW
;c
(c
m
;u
m
) : �g
(31)
for all � 2 L
1
(; IR
N
) \H
1
(; IR
N
).
(iii)
Z
W
;E
(c
m
; E(u
m
)) : r� =
Z
S
�
: r� (32)
for all � 2 H
1
(; IR
n
).
Here, w
m
= G
�
c
m
�c
m�1
�t
�
+ �
m
2 H
1
(; IR
N
) where
�
m
= �
Z
n
P
;c
(c
m
) +PW
;c
(c
m
; E(u
m
))
o
is a constant Lagrange multiplier. Furthermore, it holds Pw
m
= w
m
.
Proof. We choose � 2 L
1
(; IR
N
)\Y , � 2 X
2
and want to determine
lim
"!0
E
m;�t
(c
m
+ "�;u
m
+ "�)�E
m;�t
(c
m
;u
m
)
"
: (33)
Since
1
is convex, it holds
1
(c
m
) �
1
(c
m
+ "�)� "
1
;c
(c
m
+ "�) � �
which implies by Assumption (A3.2) that
1
(c
m
+ "�) �
1
(c
m
) + j"P
1
;c
(c
m
+ "�)jk�k
L
1
�
1
(c
m
) + j"j
1
(c
m
+ "�)k�k
L
1
+ Cj"j:
Hence, for " small we obtain
�
1
(c
m
+ "�) + jP
1
;c
(c
m
+ "�)j
�
� C
�
1
(c
m
) + 1
�
and therefore,
�
�
�
�
1
(c
m
+ "�)�
1
(c
m
)
"
�
�
�
�
� C(
1
(c
m
) + 1):
The dominated convergence theorem of Lebesgue and Assumption (A3.3)
yield
lim
"!0
1
"
�
Z
(c
m
+ "�)�(c
m
)
�
=
Z
;c
(c
m
) � �:
20
Using the growth condition (A4.3) we compute
lim
"!0
Z
1
"
(W (c
m
+ "�; E(u
m
+ "�))�W (c
m
; E(u
m
)))
=
Z
(W
;c
(c
m
; E(u
m
))� +W
;E
(c
m
; E(u
m
)) : r�) : (34)
Since the term
1
2�t
kd � c
m�1
k
2
L
is quadratic we obtain
lim
"!0
1
"
1
2�t
�
k(c
m
+ "�)� c
m�1
k
2
L
� kc
m
� c
m�1
k
2
L
�
=
�
c
m
� c
m�1
�t
; �
�
L
=
�
G
�
c
m
� c
m�1
�t
�
; �
�
L
2
:
Since
w
m
= G
�
c
m
� c
m�1
�t
�
+ �
m
and
�
m
= �
Z
(P
;c
(c
m
) +PW
;c
(c
m
; E(u
m
)))
we obtain (30) taking into account that w
m
and c
m
� c
m�1
lie on T�
almost everywhere. The identity (31) follows from the above compu-
tations for all � 2 L
1
(; IR
N
) \ Y since (33) is equal to zero for the
minimiser (c
m
;u
m
). For arbitrary � 2 L
1
(; IR
N
) \H
1
(; IR
N
) equa-
tion (31) follows because of the de�nition of �
m
and since Pw
m
= w
m
.
The identity (32) follows for all � 2 X
2
from varying E
m;�t
with
respect to u. For arbitrary � 2 H
1
(; IR
n
) we obtain (32) since W
;E
and S
�
are symmetric. ut
3.5. Uniform estimates. So far we determined solutions (c
m
;u
m
)
and related generalised chemical potential di�erencesw
m
(m = 1; :::;M)
for every �xedM 2 IN. Now we de�ne the piecewise constant extension
(c
M
;w
M
;u
M
) of (c
m
;w
m
;u
m
)
m=1;:::;M
. For t 2 ((m � 1)�t;m�t] we
set
(c
M
;w
M
;u
M
)(�; t) := (c
m
M
;w
m
M
;u
m
M
)(�) := (c
m
;w
m
;u
m
)(�):
The piecewise linear extension (�c
M
;�w
M
;�u
M
) of (c
m
;w
m
;u
m
)
m=1;:::;M
is de�ned for t = �m�t+ (1� �)(m� 1)�t with � 2 [0; 1] as
(�c
M
;�w
M
;�u
M
)(�; t) := �(c
m
M
;w
m
M
;u
m
M
)(�)+(1��)(c
m�1
M
;w
m�1
M
;u
m�1
M
)(�):
In order to derive a priori estimates we show a discrete version of the
dissipation inequality (15) for the time discrete solutions.
21
Lemma 3.3. Assume (A1){(A7). Then the following a priori esti-
mates hold.
a) For all M 2 IN and t 2 [0; T ] we have
E(c
M
(t);u
M
(t)) +
1
2
Z
t
Lrw
M
: rw
M
� E(c
0
;u
0
):
Here, u
0
is chosen to be the minimiser of v 7!
R
fW (E(v); c
0
) +W
�
(E(v))g
in the class X
2
.
b) There exists a constant C > 0 such that
sup
t2[0;T ]
�
kc
M
(t)k
H
1
()
+ ku
M
(t)k
H
1
()
� C (35)
and
sup
t2[0;T ]
Z
1
(c
M
(t)) + krw
M
k
L
2
(
T
)
� C (36)
Proof. Taking (c
m�1
M
;u
m�1
M
) as a comparison function when minimis-
ing (28) we obtain
E(c
m
M
;u
m
M
) +
1
2�t
kc
m
M
� c
m�1
M
k
2
L
� E(c
m�1
M
;u
m�1
M
):
Using this inequality iteratively and using the de�nitions of the product
(:; :)
L
and of w
m
M
gives for all M 2 IN and m 2 f1; :::;Mg
E(c
m
M
;u
m
M
) +
1
2
Z
m�t
0
(Lrw
m
M
;rw
m
M
)
L
2
� E(c
0
;u
0
):
Due to �
R
c
m
M
= m and u
m
M
2 X
2
we can use Assumptions (A2){(A7)
and the inequalities of Poincar�e and Korn to conclude (35) and (36) ut
Since�c
M
is the piecewise linear interpolant of the c
m
M
we can rewrite
(30) as
Z
@
t
�c
M
� � +
Z
Lrw
M
: r� = 0 for all � 2 H
1
(; IR
N
); (37)
which holds for almost all t 2 (0; T ). This identity together with the
estimates in Lemma 3.3 enables us to control time di�erences of�c
M
and c
M
.
Lemma 3.4. There exists a constant C > 0 such that for all t
1
; t
2
2
[0; T ]
k�c
M
(t
2
)��c
M
(t
1
)k
L
2
()
� Cjt
2
� t
1
j
1
4
:
22
Furthermore, we can choose a subsequence (c
M
)
M2N
with N � IN and
a c 2 L
1
(0; T ;H
1
()) such that for all � 2 (0;
1
4
)
(i)�c
M
! c in C
0;�
([0; T ];L
2
());
(ii) c
M
! c in L
1
(0; T ;L
2
());
(iii) c
M
! c almost everywhere in
T
;
(iv) c
M
! c weak-* in L
1
(0; T ;H
1
());
(v)
;c
(c
M
) !
;c
(c) in L
1
(
T
);
as M 2 N tends to in�nity.
Proof. We choose t
1
; t
2
2 IR with t
1
< t
2
and � =�c
M
(t
2
)��c
M
(t
1
) as
a test function in (37) and integrate from t
1
to t
2
(compare [3]). This
gives
k�c
M
(t
2
)��c
M
(t
1
)k
2
L
2
()
+
Z
t
2
t
1
Z
Lrw
M
: r(�c
M
(t
2
)��c
M
(t
1
))dt = 0:
Since the c
m
M
are uniformly bounded in H
1
(; IR
N
) we obtain that the
�c
M
are uniformly bounded in L
1
(0; T ;H
1
(; IR
N
)). Hence,
k�c
M
(t
2
)��c
M
(t
1
)k
2
L
2
()
�
� Ck�c
M
k
L
1
(0;T ;H
1
(;IR
N
))
Z
t
2
t
1
krw
m
k
L
2
()
(� )d�
� Ck�c
M
k
L
1
(0;T ;H
1
(;IR
N
))
(t
2
� t
1
)
1
2
krwk
L
2
(
T
)
:
This implies the existence of a C > 0 such that
k�c
M
(t
2
)��c
M
(t
1
)k
L
2
()
� Cjt
2
� t
1
j
1
4
(38)
for all t
1
; t
2
2 [0; T ].
Since (�c
M
)
M2IN
is uniformly bounded in L
1
(0; T ;H
1
(; IR
N
)) and
since the embedding from H
1
(; IR
N
) into L
2
(; IR
N
) is compact we
can use the equicontinuity of (�c
M
)
M2IN
(see (38)) to apply the theo-
rem of Arzel�a{Ascoli for functions with values in a Banach space (see
Theorem 8.4). Hence, there exists a c such that (possibly along a
subsequence)
�c
M
! c in C
0;�
(0; T ;L
2
())
for all � 2 (0;
1
4
).
23
Choosing for t 2 [0; T ] values m 2 f1; :::;Mg and � 2 [0; 1] such that
t = �m�t+ (1 � �)(m� 1)�t we obtain
k�c
M
(t)� c
M
(t)k
L
2= k�c
m
M
+ (1� �)c
m�1
M
� c
m
M
k
L
2
= (1� �)kc
m
M
� c
m�1
M
k
L
2
� C(�t)
1
4
which tends to zero as M tends to in�nity. Together with (i) this
implies (ii). In particular, we obtain that c
M
converges in L
2
(
T
).
Therefore, we can extract a subsequence which converges almost ev-
erywhere. This shows (iii). The existence of a subsequence ful�lling
(iv) follows from the boundedness of c
M
in L
1
(0; T ;H
1
()) (see (35)).
It remains to show (v). The continuity of
;c
and (iii) imply
;c
(c
M
)!
;c
(c)
almost everywhere. Using Assumption (A3.2) we get for all � and all
measurable sets E �
Z
E
j
1
;c
(c
M
)j � �
Z
E
1
(c
M
) + C
�
jEj
� �C + C
�
jEj:
Hence,
R
E
j
1
;c
(c
M
)j converges to zero uniformly in M as jEj tends to
zero. Employing the convergence theorem of Vitali (see [2, 101]) we
have
1
;c
(c
M
) !
1
;c
(c) in L
1
(
T
). The growth condition (A3.3) and
Lebesgue's theorem yields the convergence of
2
;c
(c
M
). This completes
the proof of the lemma. ut
We proceed in showing compactness in (u
M
)
M2IN
and (w
M
)
M2IN
. In
general E(u
M
) enters the equation for the chemical potential di�erences
quadratically. Therefore, we will need strong convergence of fru
M
g
M
in L
2
(
T
).
Lemma 3.5. There exist subsequences (u
M
)
M2N
, (w
M
)
M2N
with N �
IN and u 2 L
1
(0; T ;H
1
()), w 2 L
2
(0; T ;H
1
()) such that
u
M
! u in L
2
(0; T ;H
1
())
and
w
M
! w weakly in L
2
(0; T ;H
1
())
as M 2 N tends to in�nity,
Proof. The estimate (35) and Korn's inequality (see Theorem 8.3)
imply the existence of a subsequence such that
u
M
! u weakly in L
2
(0; T ;H
1
())
24
for a u 2 L
2
(0; T ;X
2
). Choosing � = (u
M
� u)(t) for t 2 ((m �
1)�t;m�t) as a test function in (32) yields for almost all t 2 [0; T ]
Z
(W
;E
(c
M
; E(u
M
)) : r(u
M
� u)) (t) =
Z
(S
�
: r(u
M
� u)) (t):
Integrating in time from 0 to T , taking into account the symmetry of
W
;E
and S
�
and using the monotonicity of W
;E
we can deduce
c
1
kE(u
M
� u)k
L
2
(
T
)
�
Z
T
(W
;E
(c
M
; E(u
M
))�W
;E
(c
M
; E(u))) : E(u
M
� u)
= �
Z
T
W
;E
(c
M
; E(u)) : E(u
M
� u) +
Z
T
S
�
: E(u
M
� u):
Now we can show that the right hand side converges to zero. To
conclude this we use the growth condition on W
;E
, the convergence
of fc
M
g
M2IN
in L
2
(
T
) and the weak convergence of fu
M
g
M2IN
in
L
2
(0; T ;H
1
()). Hence, kE(u
M
� u)k
L
2
(
T
)
converges to zero and
then Korn's inequality (see Theorem 8.3) implies strong convergence
of fu
M
g
M2IN
in L
2
(0; T ;H
1
()).
Estimate (36) yields a uniform bound of rw
M
in L
2
(
T
). Also we
know that for all t 2 [0; T ]
�
Z
w
M
(t) = �
M
(t) = �
Z
(P
;c
(c
m
) +PW
;c
(c
m
; E(u
m
))) :
Taking assumptions (A3) and (A4) into account we conclude that the
right hand side is uniformly bounded. By the generalised Poincar�e
inequality we infer that w
M
is uniformly bounded in L
2
(0; T ;H
1
()).
This shows the existence of a subsequence of fw
M
g
M2IN
converging
weakly in L
2
(0; T ;H
1
()).
ut
3.6. Proof of the existence theorem.
Proof of Theorem 3.1. It remains to show that the triple (c;w;u)
obtained in the previous subsection is a weak solution in the sense of
De�nition 3.1. Therefore we need to pass to the limit in (30){(32) using
the convergence properties obtained above.
Equation (37) implies that for all � 2 L
2
(0; T ;H
1
(; IR
N
)) with
@
t
� 2 L
2
(
T
) and �(T ) = 0
�
Z
T
@
t
�(�c
M
� c
0
) +
Z
T
Lrw
M
: � = 0:
25
Using the convergence properties of�c
M
and w
M
(see Lemmas 3.4 and
3.5) we obtain the �rst equality (see (19)) in the de�nition of a weak
solution.
Choosing a � 2 L
2
(0; T ;H
1
(; IR
N
))\L
1
(
T
; IR
N
) we conclude from
(31)
Z
T
w
M
� � =
Z
T
fP�rc
M
: r� +P
;c
(c
M
) � � +PW
;c
(c
M
;rE(u
M
)) � �g :
Clearly the linear terms converge to the analogous expressions in (20).
The convergence
Z
T
;c
(c
M
) � � !
Z
T
;c
(c) � �
follows with the help of the convergence theorem of Vitali by using
the growth condition on
;c
(see (A3.2)), the estimate on
1
(c
M
) in
(36), the convergence almost everywhere of c
M
and the boundedness
of �. The generalised convergence theorem of Lebesgue, the growth
assumption (A4.3) together with the strong convergence of ru
M
and
c
M
in L
2
yields that we can pass to the limit in
R
T
W
;c
(c
M
; E(u
M
)) ��.
To pass to the limit in the elasticity system (32) is straightforward
using again the strong convergence of ru
M
and c
M
in L
2
(
T
) and the
growth condition (A4.3). ut
3.7. Uniqueness for homogeneous linear elasticity. We now prove
a uniqueness theorem in the case of homogeneous linear elasticity and
under the assumption that the stress free strain varies linearly with the
concentration, i.e.
E
?
(c) =
N
X
k=1
c
k
E
?
k
; (39)
where the E
?
k
= E
?
(e
k
) are the stress free strains in the case that the
material were uniformly equal to component k. Altogether the elastic
part of the free energy has the form
W (c; E) =
1
2
(E � E
?
(c)) : C (E � E
?
(c)) (40)
with a constant positive de�nite tensor C which is assumed to ful�l the
usual symmetry conditions of linear elasticity (see Section 9). Let us
note explicitely that we do not assume that C is isotropic. This takes
26
into account that in applications C in general will be an anisotropic
tensor. Now the elastic Cahn{Hilliard system becomes
@
t
c = L�w;
w = P
�
�r � �rc+
;c
(c)� (E
?
k
: C (E(u)� E
�
(c)))
k=1;:::;N
�
;
0 = r � C (E(u) � E
?
(c)) (41)
which has to be solved together with the boundary conditions (11){(13)
and the initial condition (14).
Assume there exist two weak solutions (c
1
;w
1
;u
1
) and (c
2
;w
2
;u
2
),
which solve the elastic Cahn{Hilliard system in the sense of De�nition
3.1. Then formally it holds (compare Subsection 3.1)
1
2
d
dt
kc
2
� c
1
k
2
L
+
Z
�r(c
2
� c
1
) : r(c
2
� c
1
)
+
Z
(
;c
(c
2
)�
;c
(c
1
)) � (c
2
� c
1
)
+
Z
(E((u
2
� u
1
)� E
?
(c
2
� c
1
)) : C (E((u
2
� u
1
)� E
?
(c
2
� c
1
))
= 0:
Our goal is to use this identity together with the convexity of
1
,
suitable conditions on
2
and the inequalities of Gronwall and Korn to
prove uniqueness with respect to c and u. Then the identity (8) gives
uniqueness with respect to w.
Theorem 3.2. Assume (A1){(A3) and (A5){(A7), suppose
2
;c
is Lip-
schitz continuous and let W have the form (39), (40).
Then there exists a unique weak solution of the elastic Cahn{Hilliard
system in the sense of De�nition 3.1.
Proof. To prove the uniqueness result we generalise an idea of Blowey
and Elliott [14] to the case that elastic e�ects are included. Assume
there are two solutions (c
1
;w
1
;u
1
) and (c
2
;w
2
;u
2
) in the class
L
2
(0; T ;H
1
(; IR
N
))� L
2
(0; T ;H
1
(; IR
N
))� L
2
(0; T ;X
2
). We de�ne
c := c
2
� c
1
; w := w
2
�w
1
; u = u
2
� u
1
:
Then it holds
�
Z
T
c � @
t
� +
Z
T
Lrw : r� = 0
27
for all � 2 L
2
(0; T ;H
1
(; IR
N
)) with @
t
� 2 L
2
(
T
) and �(T ) = 0. For
t
0
2 (0; T ) and given � 2 L
2
(0; T ;H
1
(; IR
N
)) we de�ne
�(:; t) :=
(
R
t
0
t
�(:; s) ds if t � t
0
,
0 if t > t
0
.
Then we have
0 =
Z
t
0
c � � +
Z
t
0
Lrw : r
�
Z
t
0
t
�
�
=
Z
t
0
c � � +
Z
t
0
Lr
�
Z
t
0
w
�
: r�: (42)
This implies
Gc = �
Z
t
0
w and @
t
Gc = �w:
Taking � = w in (42) gives
0 =
Z
t
0
c �w +
Z
t
0
Lr (Gc) : r (@
t
Gc) :
Since the second integrand is a time derivative, we can use c(0) =
Gc(0) = 0 to obtain
0 =
Z
t
0
c �w +
Z
LrGc : rGc(t
0
): (43)
Taking the di�erence of the equation (20) for w
2
and w
1
we obtain
Z
T
w � � =
Z
T
n
�rc : r� + (
;c
(c
2
)�
;c
(c
1
)) �P�
� (E
?
k
: C (E(u) � E
�
(c)))
k=1;:::;N
�P�
o
for all � 2 L
2
(0; T ;H
1
(; IR
N
))\L
1
(
T
; IR
N
). Now we choose the test
function � = X
[0;t
0
]
P
M
(c) = X
[0;t
0
]
P
M
(c
2
� c
1
), where we de�ned
P
M
(c
0
) :=
(
c
0
if jc
0
j �M ,
c
0
jc
0
j
M if jc
0
j > M ,
for all M > 0 and all c
0
2 IR
N
. Then the inequality
�
1
;c
(c
2
)�
1
;c
(c
1
)
�
�P
M
(c
2
� c
1
) � 0
28
leads to
Z
t
0
n
�rc : rP
M
(c) +
�
2
;c
(c
2
)�
2
;c
(c
1
)
�
�P
M
(c) (44)
� (E
?
k
: C (E(u) � E
�
(c)))
k=1;:::;N
�P
M
(c)�w �P
M
(c)
o
� 0:
In the limitM !1 we obtain (44) with P
M
(c) replaced by c.
Now we choose u = (u
2
�u
1
)X
(0;t
0
)
as a test function in the di�erence
of the equations for u
2
and u
1
(see (41)), to discover
Z
t
0
C (E(u) � E
?
(c)) : E(u) = 0: (45)
Using (43), (44) and (45) we obtain
kck
L
(t
0
) +
Z
t
0
�rc : rc+
Z
t
0
(E(u) � E
?
(c)) : C (E(u) � E
?
(c))
� �
Z
t
0
�
2
;c
(c
2
)�
2
;c
(c
1
)
�
� c:
Taking into account the Lipschitz continuity of
2
;c
we can use (16) and
the Gronwall inequality to conclude
kck
L
= 0 and hence c = 0:
Then we obtain
Z
T
E(u) : CE(u) = 0
and Korn's inequality (see Theorem 8.3) implies
u = 0:
Now (20) ensures that w is uniquely de�ned. This completes the proof
of the uniqueness theorem. ut
4. Logarithmic free energy
A homogeneous free energy density derived from a mean{�eld the-
ory is the sum of a logarithmic entropy term and a pairwise interaction
term and has the form
(c) = �
N
X
k=1
c
k
ln c
k
+
1
2
c �Ac; (46)
29
where � 2 IR
+
is the temperature and the entries of the matrix A =
(A
kl
)
k;l=1;:::;N
are the constant parameters that describe pairwise in-
teractions between the components. For simplicity we have rescaled
the absolute temperature such that the Boltzmann constant k
B
is
equal to one. We assume that A is symmetric. A typical example
is A = X (ee
t
� Id) which means that the interactions between all
components have the same magnitude X 2 IR
+
. Let us point out that
phase separation only occurs if A has negative eigenvalues which im-
plies that c 7�! c �Ac is not positive de�nite. Then the second term
in (46) is non{convex while the �rst one is convex. For small temper-
atures � the homogeneous free energy then has more than one local
minima, leading to the occurrence of di�erent phases.
Energies of the form (46) have been studied for example by De
Fontaine [25, 26], Hoyt [52, 53, 54] and Elliott and Luckhaus [31]. A ho-
mogeneous free energy of the form (46) implies that
;c
, a term which
enters the equation for the chemical potential di�erences w, becomes
singular if one of the c
k
(k = 1; :::; N) tends to zero.
In the literature the logarithmic term often is approximated by a
polynomial leading to a smooth free energy density as studied in the
previous section. The system of di�erential equations for the concen-
trations introduced in Section 2 is a system of fourth order parabolic
equations and in general the concentrations can attain non{physical
negative values. In fact, a smooth does not guarantee that the con-
centrations remain non{negative. On the other hand it will turn out
that the singular term, due to the presence of the logarithmic con-
tribution to the homogenous free energy density, prevents the c
k
from
attaining negative values. It is the goal of this section to study the evo-
lution equations which result from a homogeneous free energy density
of the form (46). As already mentioned the system contains singular
terms and the analysis is much more complicated than the one in the
previous section.
The case of the Cahn{Hilliard system with logarithmic free energy
but without elasticity has been already studied by Elliott and Luck-
haus [31]. They proved an existence and uniqueness result. For their
analysis it was crucial to derive a priori estimates by di�erentiating
the equation for the chemical potential di�erences w (see equation (8))
with respect to time. Since in general the solutions of the elastic Cahn{
Hilliard system will not be smooth enough to allow di�erentiating with
respect to time, we had to develop a new method. Let us mention
one main di�culty and the technique to overcome this di�culty. The
strain tensor enters the equation of the chemical potential di�erences
quadratically. Hence, we need to establish a higher integrability result
30
for the strain. We will apply the technique of Modica and Giaquinta
[46] (see also [45]), who derived for solutions to elliptic systems a higher
integrability result. This is the base to show that ln c
i
, and hence
;c
lie in L
q
(
T
) (for some q > 1). In conclusion we can derive that the
chemical potentials are well de�ned. In particular, we can also show
that the concentrations c
i
are positive almost everywhere.
Let us state the setting in which we study the resulting system of
equations. As in the previous section we want to solve the system of
equations (7){(10) with the initial and boundary conditions (11){(14).
We show existence of a weak solution in the sense of De�nition 3.1.
As pointed out above the homogeneous free energy (46) will guarantee
that solutions remain non{negative. Hence, we seek solutions that lie
on the Gibbs{simplex
G := fc
0
2 IR
N
j
N
X
i=1
c
0
k
= 1 and c
0
k
� 0 for k = 1; :::; Ng
containing all vectors of concentrations which are physically meaning-
ful.
For the existence theory we will require that the Assumptions (A1)
and (A4){(A6) from the previous section hold. The Assumptions (A2),
(A3) and (A7) will be replaced by the following ones.
To simplify the presentation we require that the gradient energy
tensor � ful�ls
(A2
0
) � = Id with 2 IR
+
.
Furthermore, we assume
(A3
0
) is of the form (46) with � 2 IR
+
and A 2 IR
N�N
is symmetric
and positive de�nite,
(A7
0
) the initial data c
0
2 X
1
ful�l c
0
2 G almost everywhere and
�
Z
c
0
k
> 0 for k = 1; :::; N:
The assumption on the mean value of c does not give any practical
limitation. A zero mean value for one component, i.e. �
R
c
0
k
= 0,
together with c
0
2 G would imply that c
k
= 0 almost everywhere.
Therefore, component k does not appear at all, which means that we
can reduce the system to a N � 1 system containing all components
beside the k{th component.
The main result of this section is an existence result for the elastic
Cahn{Hilliard system with a logarithmic free energy.
31
Theorem 4.1. (Existence) Assume (A1), (A2
0
), (A3
0
), (A4){(A6)
and (A7
0
). Then there exists a weak solution of the elastic Cahn{
Hilliard system in the sense of De�nition 3.1 which has the following
properties:
(i) c 2 C
0;
1
4
([0; T ];L
2
());
(ii) @
t
c 2 L
2
�
0; T ; (H
1
())
�
�
,
(iii) there exists a p > 2 such that u 2 L
1
(0; T ;W
1;p
(; IR
n
)) ;
(iv) there exists a q > 1 such that for k 2 f1; :::; Ng
ln c
k
2 L
q
(
T
):
In particular, c
k
> 0 almost everywhere.
As in the case of a smooth energy we have a uniqueness theorem if
the elastic free energy density is homogeneous and quadratic in E.
Theorem 4.2. (Uniqueness) In addition to the assumptions of The-
orem 4.1 we assume that W has the form (39), (40).
Then there exists a unique solution of the elastic Cahn{Hilliard sys-
tem with logarithmic free energy in the sense of De�nition 3.1.
The uniqueness theorem is proved in exactly the same way as in the
case of a smooth homogeneous free energy (see the proof of Theorem
3.2).
4.1. A regularised problem. Our goal is to approximate the singular
system by a system with smooth free energies such that the theory of
the previous section can be applied.
First of all we assume that the elastic free energy density W ful�ls
(A4.4) W
;c
(c
0
; E
0
) = 0 for all c
0
2 IR
N
with jc
0
j > 2 and all E
0
2 IR
n�n
.
This assumption is without loss of generality, because the solution
turns out to lie on the Gibbs simplex and therefore has modulus less
than two.
Furthermore, for given � > 0 we replace by the C
2
{function
�
(c
0
) = �
N
X
k=1
�
(c
0
k
) +
1
2
c
0
�Ac
0
(47)
with
�
(d) :=
�
d ln d for d � � ;
(d ln � �
�
2
+
d
2
2�
) for d < �:
(48)
For later use we de�ne
1;�
(c
0
) = �
N
X
k=1
�
(c
0
k
) and
2
(c
0
) =
1
2
c
0
�Ac
0
:
32
The same regularisation has been used by Elliott and Luckhaus [31] in
their existence proof for the Cahn{Hilliard system without elasticity.
The following lemma (for a proof see Elliott and Luckhaus [31]) states
that
�
is uniformly bounded from below on �.
Lemma 4.1. There exist a �
0
> 0 and a K > 0 such that for all
� 2 (0; �
0
)
�
(c
0
) � �K for all c
0
2 �:
For the rest of this section we shall assume (A1), (A2
0
), (A4), (A4.4),
(A5), (A6) and (A7
0
). The following lemma states an existence result
for the regularised problem and collects a priori estimates and com-
pactness results, which can be obtained similar as in Section 3.
Lemma 4.2. Suppose the homogeneous free energy density is of the
form (47).
(a) For all � 2 (0; �
0
) there exists a weak solution (c
�
;w
�
;u
�
) of the
elastic Cahn{Hilliard system in the sense of De�nition 3.1.
(b) Moreover, there exists a constant C > 0 such that for all � 2
(0; �
0
)
sup
t2[0;T ]
�
kc
�
(t)k
H
1
()
+ ku
�
(t)k
H
1
()
� C;
sup
t2[0;T ]
Z
1;�
(c
�
(t)) + krw
�
k
L
2
(
T
)
� C
and
kc
�
(t
2
)� c
�
(t
1
)k
L
2
()
� Cjt
2
� t
1
j
1
4
for all t
1
; t
2
2 [0; T ].
(c) Furthermore, one can extract a subsequence (c
�
)
�2R
, where R �
(0; �
0
) is a countable set with zero as the only cluster point, such that
(i) c
�
! c in C
0;�
([0; T ];L
2
()) for all � 2 (0;
1
4
);
(ii) c
�
! c almost everywhere;
(iii) c
�
! c weak-* in L
1
(0; T ;H
1
());
(iv) u
�
! u in L
2
(0; T ;H
1
());
as � 2 R tends to zero.
Proof. The regularised problem ful�ls the assumptions of Theorem
3.1 for � 2 (0; �
0
). To show this, one also makes use of Lemma 4.1.
Hence, a weak solution of the regularised problem exists. The a priori
estimates in (b) follow from the Lemmas 3.3 and 3.4 by convergence
and lower semi-continuity properties. To show that the constant on the
right hand side does not depend on �, one has to check that E
�
(c
0
;u
0
)
does not depend on � (see the proof of Lemma 3.3). This is implied
33
by the facts that the initial data c
0
lie in H
1
(; IR
N
) and only attain
values on the Gibbs simplex. The convergence properties in (c) follow
as in the proofs of the Lemmas 3.4 and 3.5. ut
What remains to be done? It is our goal to show compactness
for the chemical potential di�erences fw
�
g
�2(0;�
0
)
. We already estab-
lished a uniform estimate for frw
�
g
�2(0;�
0
)
, i.e. it is enough to con-
trol the spatial mean values of fw
�
g
�2(0;�
0
)
to get a uniform bound
in L
2
(0; T ;H
1
(; IR
N
)). This will be our �rst step. Thereafter, it is
possible to show the existence of a subsequence of fw
�
g
�2(0;�
0
)
which
converges weakly in L
2
(0; T ;H
1
(; IR
N
)) to a limit w. Then it re-
mains to prove that the following equation holds in a weak sense (see
De�nition 3.1)
w = P (� �u+
;c
(c) +W
;c
(c; E(u))) ; (49)
where
;c
(c) = � (ln c
k
+ 1)
k=1;:::;N
+Ac:
The problem is that ln c
k
might be singular. Our goal is to establish a
uniform estimate for
�
�
�
0
(c
�
k
) in L
q
(
T
) for some q > 1. We remind
the reader that
�
is an approximation of (d) = d ln d. To show the
L
q
{bound we �rst derive the integrability of E(u) =
1
2
(ru+ (ru)
t
) in
L
p
(
T
) for some p > 2. This implies that W
;c
lies in L
p
2
(
T
) which
allows to multiply the equations in (49) by an appropriate power of
�
�
�
0
(c
�
k
), leading to uniform L
q
{bounds (for some q > 1) for
�
�
�
0
(c
�
k
).
The uniform estimates for
�
�
�
0
(c
�
k
) together with the almost every-
where convergence of fc
�
g
�2(0;�
0
)
yields the convergence of
�
�
�
0
(c
�
k
) to
ln c
k
+1 in L
1
(
T
). This is enough to pass to the limit in the equation
for w
�
and to show that c
k
> 0 almost everywhere for k = 1; :::; N .
As pointed out above we �rst have to derive a uniform bound on
fw
�
g
�2(0;�
0
)
:
Lemma 4.3. (i) There exists a constant C > 0 independent of � such
that for all � 2 (0; �
0
)
Z
T
0
�
�
Z
P
1;�
;c
(c
�
)
�
2
(t) dt < C
and
kw
�
k
L
2
(0;T ;H
1
())
< C:
(ii) There exists a subsequence (w
�
)
�2R
where R � (0; �
0
) is a countable
set with zero as the only cluster point such that
w
�
! w weakly in L
2
(0; T ;H
1
()):
34
Proof. We de�ne
w
�
0
= w
�
� �
�
with
�
�
= �
Z
w
�
= �
Z
�
P
�
;c
(c
�
) +PW
;c
(c
�
; E(u
�
))
�
:
To derive a bound on the Lagrange multipliers �
�
we generalise an
idea of Barrett and Blowey [11]. Since w
�
is a solution in the sense of
De�nition 3.1 with homogeneous free energy density
�
we have:
Z
(w
�
0
+ �
�
) � � = (50)
Z
�
rc
�
: r� +P
�
;c
(c
�
) � � +PW
;c
(c
�
; E(u
�
)) � �
for all � 2 H
1
(; IR
N
) \ L
1
(; IR
N
) and for almost all t 2 (0; T ).
For all elements k lying on the Gibbs simplexG we obtain by using
the convexity of
1;�
and the fact that k� c
�
2 T� almost everywhere
Z
1;�
(k) �
Z
1;�
(c
�
) +
Z
1;�
;c
(c
�
) � (k� c
�
)
=
Z
1;�
(c
�
) +
Z
P
1;�
;c
(c
�
) � (k� c
�
): (51)
SinceW ful�lsW
;c
(c
0
; E
0
) = 0 for jc
0
j � 2, we can also choose � = k�c
�
as a test function in (50) for almost all t 2 (0; T ). Taking the resulting
expression and using inequality (51) we conclude
Z
1;�
(k) �
Z
1;�
(c
�
)�
Z
P
2
;c
(c) � (k� c
�
)
�
Z
PW
;c
� (k� c
�
) +
Z
rc
�
: rc
�
+
Z
w
�
0
� (k� c
�
) +
Z
�
�
� (k� c
�
)
for almost all t 2 (0; T ). We want to use the above inequality to
establish an estimate of the term
Z
�
�
� (k� c
�
):
Using
jPW
;c
(c
0
; E
0
) � (k� c
0
)j �
(
C
2
(jE
0
j
2
+ 1) if jc
0
j < 2,
0 if jc
0
j � 2,
35
Lemma 4.1, the a priori estimates of Lemma 4.2 and Poincar�e's in-
equality for functions with mean value zero, we obtain for almost all
t 2 (0; T )
Z
�
�
� (k� c
�
) � C
�
1 + krw
�
(t)k
L
2
()
�
1 + kc
�
(t)k
L
2
()
�
+
+kc
�
(t)k
2
L
2
()
+ kru
�
k
2
L
2
()
�
: (52)
Assumption (A7
0
) and the fact that
R
c
�
(t) is constant in time ensures
the existence of a � > 0 such that for all k 2 f1; :::; Ng and all t 2 (0; T ]
� < �
Z
c
�
k
(t) < 1� �:
Choosing
k = �
Z
c
�
(t) + � sign(�
�
k
� �
�
l
) (e
k
� e
l
) 2 G
in (52) gives
j�
�
k
� �
�
l
j(t) �
C
�jj
�
1 + krw
�
(t)k
L
2
()
�
:
Integrating j�
�
k
� �
�
l
j
2
(t) from 0 to T and using the identity �
�
=
1
N
�
P
N
l=1
(�
�
k
� �
�
l
)
�
k=1;:::;N
leads to
Z
T
0
j�
�
j
2
(t) dt � C:
This, together with the growth condition for W and the a priori esti-
mates of Lemma 4.2, gives an estimate for the spatial mean values of
w
�
in L
2
(0; T ). Hence, the Poincar�e inequality yields the second in-
equality in (i). The second hypothesis then follows from a compactness
argument. ut
4.2. Higher integrability for the strain tensor. In this subsection
we use a perturbation argument to show that the deformation gradient
has the integrability property:
there exists a p > 2 such that for almost all t 2 [0; T ] we have ru(t) 2
L
p
().
Lemma 4.4. (Higher integrability: interior estimates) Suppose
that c 2 L
�
(; IR
N
), � > 2 and that (A4) and (A5) hold.
36
Then there exists a p 2 (2; �], independent of c, such that for all
u 2 H
1
(; IR
n
) which ful�l for all � 2 H
1
(; IR
n
) the identity
Z
W
;E
(c; E(u)) : r� =
Z
S
�
: r� (53)
the integrability property
ru 2 L
p
loc
(; IR
n�n
)
holds. In particular, for all
0
�� it holds
kruk
L
p
(
0
;IR
n�n
)
� C
�
kruk
L
2
(;IR
n�n
)
+ kck
L
p
(;IR
n
)
+ 1
�
where C = C(;
0
; C
2
; c
2
; c
1
;S
�
; n; �; p) is independent of c.
Proof. The proof is based on a Caccioppoli inequality, a reverse
H�older inequality and a perturbation argument due to Gehring [44]
and Giaquinta and Modica [46]. This technique is well known for el-
liptic systems. In our case additional di�culties arise in the derivation
of the Caccioppoli inequality because the direct estimates only con-
trol E(u) rather than ru. Therefore, we present the derivation of the
Caccioppoli inequality in detail.
Let x
0
2 and R > 0 be such that
Q
2R
(x
0
) := fx 2 IR
n
j jx
i
� x
0i
j < 2Rg � :
Then we de�ne a cuto� function � 2 C
1
0
() with the properties
i) � = 0 in nQ
2R
(x
0
),
ii) 0 � � � 1 in and � = 1 in Q
R
(x
0
),
iii) jr�j �
2
R
.
Now we want to test equation (53) with
� = �
2
(u� �) with � 2 IR
n
:
We compute
E(�) = �
2
E(u) + �
�
(u� �) (r�)
t
+r� (u� �)
t
�
:
Due to the symmetry of W
;E
(c; E(u)) we obtain
Z
�
2
W
;E
(c; E(u)) : E(u) + 2
Z
�W
;E
(c; E(u)) :
�
(u� �) (r�)
t
�
=
Z
�
2
S
�
: E(u) + 2
Z
�S
�
:
�
(u� �) (r�)
t
�
: (54)
Assumptions (A4.2) and (A4.3) yield
c
1
jE(u)j
2
� W
;E
(c; E(u)) : E(u) + C
2
(jcj+ 1) jE(u)j
and
jW
;E
(c; E(u)) :
�
(u� �) (r�)
t
�
j � C
2
(jE(u) + jcj+ 1) ju� �j
2
R
:
37
Since S
�
is a constant tensor and using Young's inequality we can
deduce from (54) the existence of a constant C > 0 depending on
c
1
; C
2
and jS
�
j such that
c
1
Z
�
2
jE(u)j
2
�
� C
Z
�
2
�
jcj
2
+ 1
�
+ C
Z
� (jE(u)j + jcj+ 1) ju� �j
2
R
� C
Z
�
2
�
jcj
2
+ 1
�
+
C
R
2
Z
Q
2R
(x
0
)
ju� �j
2
:
Employing
E(�(u � �)) = �E(u) +
1
2
�
(u� �)(r�)
t
+r�(u� �)
t
�
we obtain
Z
jE (�(u� �)) j
2
� 2
�
Z
�
2
jE(u)j
2
+
Z
ju� �j
2
jr�j
2
�
:
Now we can apply Korn's inequality for functions with boundary value
zero to conclude
Z
jr (�(u� �)) j
2
� C
Z
�
2
�
jcj
2
+ 1
�
+
C
R
2
Z
Q
2R
(x
0
)
ju� �j
2
: (55)
Since
r (�(u� �)) = �ru+ (u� �) (r�)
t
we derive from (55) that
�
Z
Q
R
(x
0
)
jruj
2
� C �
Z
Q
2R
(x
0
)
�
jcj
2
+ 1
�
+
C
R
2
�
Z
Q
2R
(x
0
)
ju� �j
2
:
Now we choose � = �
R
Q
2R
(x
0
)
u and use the Poincar�e{Sobolev inequality
(see Theorem 8.1) to conclude
�
Z
Q
R
(x
0
)
jruj
2
� C �
Z
Q
2R
(x
0
)
�
jcj
2
+ 1
�
+ C
�
�
Z
Q
2R
(x
0
)
jruj
2n
n+2
�
n+2
n
:
Finally, Proposition 8.1 with g = jruj
2n
n+2
, q =
n+2
n
and f = C (jcj
2
+ 1)
n
n+2
and a covering argument leads to the assertion. ut
Theorem 4.3. (Higher integrability) Suppose that c 2 L
�
(; IR
N
),
� > 2 and that (A4) and (A5) hold.
Then there exists a p 2 (2; �], independent of c, such that for all
u 2 H
1
(; IR
n
), which ful�l for all � 2 H
1
(; IR
n
) the identity
Z
W
;E
(c; E(u)) : r� =
Z
S
�
: r� (56)
38
the integrability property
ru 2 L
p
(; IR
n�n
)
holds. In particular,
kruk
L
p
(;IR
n�n
)
� C
�
kruk
L
2
(;IR
n�n
)
+ kck
L
p
(;IR
n
)
+ 1
�
where C = C(; C
2
; c
2
; c
1
;S
�
; n; �; p) is independent of c.
Proof. The integrability in the interior follows from Lemma 4.4. Hence,
it remains to show the higher integrability at the boundary. Since
has a Lipschitz boundary, there exist for all x
0
2 @ a Lipschitz func-
tion h : IR
n�1
! IR such that { upon relabelling and reorientation of
the interface if necessary { the boundary @ locally around x
0
is the
graph of h. In addition, h can be chosen such that locally lies on one
side of the graph. To state this property precisely we de�ne the sets
Q := fy 2 IR
n
j jy
i
j < R
0
for i = 1; :::; n g;
Q
+
:= fy 2 Q j y
n
> 0 g;
Q
�
:= fy 2 Q j y
n
< 0 g;
Q
0
:= fy 2 Q j y
n
= 0 g
and the transformation
� : Q �! IR
n
;
y 7�! � (y) := (y
1
; ::::; y
n�1
; h(y
1
; ::::; y
n�1
) + y
n
):
Then we require that there exists a R
0
> 0 such that
�
�
Q
+
�
� ;
�
�
Q
�
�
� IR
n
n
�
:
In what follows, we assume that x
0
, h and R
0
are chosen such that the
above requirements hold. Now we de�ne
v : Q
+
! IR
n
and d : Q
+
! IR
n
via
v = u � � and d = c � � :
In addition, we set for all y 2 Q
g(y) =
(
jrvj
2n
n+2
(y) if y 2 Q
+
;
0 if y 2 Q nQ
+
:
Our goal is to apply Proposition 8.1 for the function g. This then shows
higher integrability of rv and by transformation also for ru.
39
Claim: There are constants b; C > 0 such that for all y
0
2 Q and all
R > 0 with 2R < dist(y
0
; @Q)
�
Z
Q
R
(y
0
)
g
q
dy � b
�
�
Z
Q
2R
(y
0
)
g dy
�
q
+�
Z
Q
2R
(y
0
)
f
q
dy (57)
where q =
n+2
n
and f = C (jdj
2
+ 1)
n
n+2
.
To prove the claim we choose a y
0
2 Q and a R <
1
2
dist(y
0
; @Q).
Then there are three possibilities:
Case 1. Q3
2
R
(y
0
) \Q
+
= ;.
The left hand side in (57) in this case is zero and hence the inequality
holds.
Case 2. Q3
2
R
(y
0
) \Q
�
= ;.
Denoting by L the Lipschitz constant of h, it holds that � (Q
R
(y
0
))
and � (Q3
2
R
(y
0
)) have a distance larger than
R
4
min(1;
1
L
). Hence, we
can choose a cuto� function � 2 C
1
0
() with the properties
i) � = 0 in n �
�
(Q3
2
R
(y
0
)
�
,
ii) 0 � � � 1 in and � = 1 in � (Q
R
(y
0
)),
iii) jr�j �
8
R
max(1; L).
Testing (56) with � = �
2
(u � �) where � 2 IR
n
and concluding as
in Lemma 4.4 we obtain
Z
� (Q
R
(y
0
))
jruj
2
� C
Z
�
�
Q
3
2
R
(y
0
)
�
�
jcj
2
+ 1
�
+
C
R
2
Z
�
�
Q
3
2
R
(y
0
)
�
ju��j
2
:
Transforming the integrals leads to
Z
Q
R
(y
0
)
jrvj
2
� C
Z
Q
3
2
R
(y
0
)
�
jdj
2
+ 1
�
+
C
R
2
Z
Q
3
2
R
(y
0
)
jv ��j
2
where C depends on L. Choosing � = �
R
Q
3
2
R
(y
0
)
v and using the
Sobolev{Poincar�e inequality we deduce
Z
Q
R
(y
0
)
jrvj
2
� C
Z
Q
3
2
R
(y
0
)
�
jdj
2
+ 1
�
+
C
R
2
0
@
Z
Q
3
2
R
(y
0
)
jrvj
2n
n+2
1
A
n+2
n
:
This implies
Z
Q
R
(y
0
)
g
q
dy �
C
R
2
�
Z
Q
2R
(y
0
)
g dy
�
q
+
Z
Q
2R
(y
0
)
f
q
dy: (58)
40
Multiplying by R
�n
now gives the result.
Case 3.
Q3
2
R
(y
0
) \Q
+
6= ; and Q3
2
R
(y
0
) \Q
�
6= ;: (59)
For all
�
R > 0 we de�ne
Q
+
�
R
(y
0
) := Q
�
R
(y
0
) \Q
+
and Q
�
�
R
(y
0
) := Q
�
R
(y
0
) \Q
�
:
From (59) it is seen that
Q
2R
(y
0
) \ Q
0
6= ;:
Hence, � (Q
2R
(y
0
)) intersects the boundary of , i.e.
� (Q
2R
(y
0
)) \ @ 6= ;:
The Lipschitz continuity of h guarantees
dist
�
�
�
@Q
+
2R
(y
0
)
�
\ ; �
�
@Q
+
R
(y
0
)
�
\
�
R
2
min
�
1;
1
L
�
which hence allows us to choose a cuto� function � 2 C
1
() with the
properties
i) � = 0 in n � ((Q
2R
(y
0
)),
ii) 0 � � � 1 in and � = 1 in � (Q
R
(y
0
) \ ),
iii) jr�j �
4
R
max(1; L).
Since �(u � �) = 0, where � 2 IR
n
, on an open part of @, Korn's
inequality holds for �(u � �) = 0. Therefore, by testing (56) with
� = �
2
(u� �) we can proceed as in Case 2 to obtain
Z
Q
+
R
(y
0
)
jrvj
2
�
Z
Q
+
2R
(y
0
)
C
�
jdj
2
+ 1
�
+
C
R
2
Z
Q
+
2R
(y
0
)
jv� �j
2
:
From (59) we conclude
L
n
�
Q
+
2R
(y
0
)
�
� cR
n
and
diamQ
+
2R
(y
0
) � CR:
With the help of the Sobolev{Poincar�e inequality we deduce
Z
Q
+
R
(y
0
)
jrvj
2
�
Z
Q
+
2R
(y
0
)
C
�
jdj
2
+ 1
�
+ C
�
L
n
�
Q
+
2R
(y
0
)
��
�
2
n
Z
Q
+
2R
(y
0
)
jrvj
2n
n+2
!
n+2
n
41
and hence
Z
Q
+
R
(y
0
)
jrvj
2
�
Z
Q
+
2R
(y
0
)
C
�
jdj
2
+ 1
�
+
C
R
2
Z
Q
+
2R
(y
0
)
jrvj
2n
n+2
!
n+2
n
:
If we integrate over the larger sets Q
R
(y
0
) and Q
2R
(y
0
) respectively we
obtain (58). Hence, as before we multiply by R
�n
and deduce (57).
Now (57) and Proposition 8.1 give the higher integrability at the
boundary. The higher integrability at the boundary, Lemma 4.4 and a
covering argument imply the desired conclusion. ut
4.3. Higher integrability for the logarithmic free energy. The
equation for the chemical potential di�erences w
�
of the regularised
system is
w
�
= � �c
�
+ �P
�
�
�
(c
�
k
)
�
k=1;:::;N
+PAc
�
+PW
;c
(c
�
; E(u
�
)); (60)
where we de�ne �
�
:=
�
�
�
0
. The function
�
is an approximation of
(d) = d ln d and hence �
�
=
�
�
�
0
becomes singular as � ! 0. We
remark that
�
was chosen such that �
�
is monotone (see (48)). This
is crucial in order to show that �
�
(c
�
k
) is uniformly bounded in L
q
(
T
)
for some q > 1. We will achieve this by testing the weak formulation
of the equation for w
k
with an appropriate power of �
�
(c
�
k
).
Lemma 4.5. There exist constants q > 1 and C > 0 such that for all
� 2 (0;min(
1
N
; �
0
)) and all k 2 f1; :::; Ng
k�
�
(c
�
k
)k
L
q
(
T
)
� C:
Proof. Let r > 0. Then we de�ne
�
�
r
(d) = �
�
(d)j�
�
(d)j
r�1
where �
�
r
is de�ned to be zero if �
�
(d) is zero, and hence �
�
r
is continuous
on IR. For r 2 (0; 1) the function �
�
r
is not di�erentiable at the zero of
the function �
�
. Hence, for � > 0 we de�ne a monotone C
1
function
�
�;�
r
which equals �
�
r
on IR n [0; 1] and which converges to �
�
r
in C(IR)
as �! 0.
The weak formulation of the equation for the chemical potential dif-
ferences w
�
is
Z
T
w
�
� � =
Z
T
n
rc
�
: r� + �P
�
�
�
(c
�
k
)
�
k=1;:::;N
� � (61)
+PAc
�
� � +PW
;c
(c
�
; E(u
�
)) � �
o
with � 2 L
2
(0; T ;H
1
(; IR
N
)) \ L
1
(
T
; IR
N
). The Sobolev embed-
ding theorem and the estimates of Lemma 4.2 imply that c
�
lies in
42
L
1
(0; T ;L
2n
n�2
()) if n � 3, in L
1
(0; T ;L
s
()) for all s 2 [1;1) if
n = 2 and in L
1
(
T
) if n = 1. We can deduce from the higher
integrability result from the last subsection (see Theorem 4.3) that
ru
�
2 L
1
(0; T ;L
p
()) (for some p > 2). We choose p such that
p 2 (2; 4] and such that in addition p 2
�
2;
2n
n�2
�
if n � 3. Hence,
W
;c
(c
�
; E(u
�
)) 2 L
1
(0; T ;L
p
2
()). This implies that also test func-
tions � 2 L
2
(0; T ;H
1
(; IR
N
)) \ L
p
p�2
(
T
; IR
N
) are allowed in (61).
We test (61) with the function
� =
�
�
�;�
r
(c
�
k
)
�
k=1;:::N
;
which is admissible for all r 2 (0; 1] with r
p
p�2
� 2 (note that �
�
(d) is
sub-linear in d). We obtain
Z
T
N
X
k=1
w
�
k
� �
�;�
r
(c
�
k
) = (62)
Z
T
N
X
k=1
(
rc
�
k
� r�
�;�
r
(c
�
k
) + �
"
�
�
(c
�
k
)�
1
N
N
X
l=1
�
�
(c
�
l
)
!#
�
�;�
r
(c
�
k
)
)
+
Z
T
n
PAc
�
�
�
�
�;�
r
(c
�
k
)
�
k=1;:::N
+PW
;c
(c
�
; E(u
�
)) �
�
�
�;�
r
(c
�
k
)
�
k=1;:::N
o
:
Furthermore, we have
N
X
k=1
"
�
�
(c
�
k
)�
1
N
N
X
l=1
�
�
(c
�
l
)
!#
�
�;�
r
(c
�
k
)
=
1
N
N
X
k;l=1
�
�
�
(c
�
k
)� �
�
(c
�
l
)
�
�
�;�
r
(c
�
k
)
=
1
N
N
X
k<l
�
�
�
(c
�
k
) � �
�
(c
�
l
)
�
�
�;�
r
(c
�
k
) +
1
N
N
X
k>l
�
�
�
(c
�
k
)� �
�
(c
�
l
)
�
�
�;�
r
(c
�
k
)
=
1
N
N
X
k<l
�
�
�
(c
�
k
) � �
�
(c
�
l
)
� �
�
�;�
r
(c
�
k
)� �
�;�
r
(c
�
l
)
�
� 0
since both �
�
and �
�;�
r
are monotone increasing.
43
Using that
�
�
�;�
r
�
0
� 0 we conclude that the �rst term on the right
hand side of (62) is non{negative. Hence, (62) implies
�
Z
T
1
N
N
X
k<l
�
�
�
(c
�
k
)� �
�
(c
�
l
)
� �
�
�;�
r
(c
�
k
)� �
�;�
r
(c
�
l
)
�
�
�
Z
T
(
N
X
k=1
w
�
k
� �
�;�
r
(c
�
k
)�
PAc
�
�
�
�
�;�
r
(c
�
k
)
�
k=1;:::N
�PW
;c
(c
�
; E(u
�
)) �
�
�
�;�
r
(c
�
k
)
�
k=1;:::N
)
� C max
k=1;:::;N
k�
�;�
r
(c
�
k
)k
L
2
(
T
)
�
kw
�
k
L
2
(
T
)
+ kc
�
k
L
2
(
T
)
�
+C
�
Z
T
jW
;c
(c
�
; E(u
�
))j
p
2
�
2
p
�
max
k=1;:::;N
Z
T
j�
�;�
r
(c
�
k
)j
p
p�2
�
1�
2
p
:
Passing to the limit � & 0 and using Theorem 4.3, the a priori esti-
mates of Lemma 4.2 and Lemma 4.3 and the inequalities of H�older and
Young proves that there exists for all � > 0 a constant C
�
such that
�
Z
T
1
N
N
X
k<l
�
�
�
(c
�
k
)� �
�
(c
�
l
)
� �
�
�
r
(c
�
k
)� �
�
r
(c
l
)
�
� (63)
� �
�
max
k=1;:::;N
Z
T
j�
�
r
(c
�
k
)j
p
p�2
�
+ C
�
:
Moreover, since
P
N
k=1
c
�
k
= 1, we have
N
min
k=1
c
�
k
�
1
N
�
N
max
k=1
c
�
k
:
44
Using this, the fact that �
�
and �
�
r
are monotone increasing functions
and Young's inequality we deduce
Z
T
N
X
k<l
�
�
�
(c
�
k
)� �
�
(c
�
l
)
� �
�
�
r
(c
�
k
)� �
�
r
(c
l
)
�
�
�
Z
T
N
max
k=1
j�
�
(c
�
k
)� �
�
(
1
N
)jj�
�
r
(c
�
k
)� �
�
r
(
1
N
)j
�
Z
T
N
max
k=1
�
�
�
�
j�
�
(c
�
k
)j
r+1
� �
�
(
1
N
)�
�
r
(c
�
k
)� �
�
(c
�
k
)�
�
r
(
1
N
) + j�
�
(
1
N
)j
r+1
�
�
�
�
�
Z
T
N
max
k=1
�
j�
�
(c
�
k
)j
r+1
� j�
�
(
1
N
)jj�
�
(c
�
k
)j
r
� j�
�
(c
�
k
)jj�
�
(
1
N
)j
r
+ j�
�
(
1
N
)j
r+1
�
�
1
2
Z
T
N
max
k=1
j�
�
(c
�
k
)j
r+1
� C:
If � <
1
N
then �
�
(
1
N
) = �(
1
N
), which ensures that the constant C is
independ of �. Together with (63) we have
Z
T
N
max
k=1
j�
�
(c
�
k
)j
r+1
�
N
2�
�
�
max
k=1;:::;N
Z
T
j�
�
(c
�
k
)j
p
p�2
r
�
+ C
�
:
Setting r =
p�2
2
and choosing � small enough gives the result.
ut
4.4. Proof of the existence theorem.
Proof of Theorem 4.1. We need to show that the limit (c;w;u)
obtained in Lemma 4.2 and Lemma 4.3 solves the elastic Cahn{Hilliard
system with logarithmic free energy. To pass to the limit in the weak
formulations of the equations
@
t
c
�
= L�w
�
and r �
�
W
;E
(c
�
; E(u
�
))
�
= 0
one can apply the same arguments as in the proof of Theorem 3.1 using
the convergence properties of (c
�
;w
�
;u
�
) and the growth condition on
W
;E
. It remains to pass to the limit in
w
�
= � �c
�
+ �P
�
�
�
(c
�
k
)
�
k=1;:::;N
+PAc
�
+PW
;c
(c
�
; E(u
�
)): (64)
Except for the term �
�
(c
�
k
) one can use similar arguments as in the
proof of Theorem 3.1 (see De�nition 3.1).
Let us show that �
�
(c
�
k
) converges almost everywhere to �(c
k
) and
that c
k
> 0 almost everywhere. Using the convergence a.e. of c
�
k
to c
k
,
the Fatou lemma and Lemma 4.5 we obtain
Z
T
lim inf
�!0
j�
�
(c
�
k
)j
q
� lim inf
�!0
Z
T
j�
�
(c
�
k
)j
q
� C:
45
Next we prove that
lim
�!0
�
�
(c
�
k
) =
(
�(c
k
) if lim
�!0
c
�
k
= c
k
> 0,
1 elsewhere
(65)
almost everywhere. First we take (x; t) 2
T
with lim
�!0
c
�
k
(x; t) =
c
k
(x; t) > 0. Since �
�
(d) =
�
�
�
0
(d) =
0
(d) = �(d) for d � � we
obtain �
�
(c
�
k
(x; t))! �(c
k
(x; t)). Now assume that (x; t) 2
T
is such
that lim
�!0
c
�
k
(x; t) = c
k
(x; t) � 0. Then we obtain for � small enough
j�
�
(c
�
k
(x; t))j � �(max
�
c
�
k
(x; t); �
�
:
The right hand side converges to 1 as � tends to zero which proves
(65). Using (65) and Lemma 4.5, we obtain
c
k
> 0 almost everywhere;
Z
T
j�(c
k
)j
q
� C
and
�
�
(c
�
k
)! �(c
k
) almost everywhere.
Since q > 1 we conclude with Vitali's theorem
�
�
(c
�
k
)! �(c
k
) in L
1
(
T
):
This is enough to pass to the limit in the weak formulation of (64).
ut
5. The sharp interface limit
In this section we study solutions of the variational problems
(P
"
) Find a minimiser (c;u) 2 X
1
�X
2
of
E
"
(c;u) :=
Z
�
"jrcj
2
+
1
"
(c) +W (c; E(u))
�
; " > 0;
subject to the constraint �
R
c =m, with m 2 �.
Solutions of the variational problems are stable stationary solutions to
the Cahn{Hilliard system with elasticity studied in Sections 3 and 4.
Under some natural assumptions it will turn out that minimisers of the
above variational problem are, roughly speaking, of the following form.
In most of the solution is close to values that minimise and the
regions where the solution is close to minimisers of are separated by
46
transition layers which are of a thickness proportional to ". It is the goal
of this section to study the limiting behaviour of E
"
and its minimisers
as " tends to zero. The scaling in " (i.e. ! "
2
,W ! "W , E !
1
"
E) is
motivated by former studies for the case when no elastic contributions
are present and by formally matched asymptotic expansions by Leo,
Lowengrub and Jou [65]. As in the case without elasticity we will use
arguments of �{convergence theory to identify the asymptotic limit for
the functionals E
"
.
For the rest of this section we assume that the homogeneous free
energy is such that
� 0 and (c
0
) = 0 , c
0
2 fp
1
; :::;p
M
g; (66)
where p
1
; :::;p
M
2 � are mutually di�erent and M � 2. We note that
the variational problem (P
"
) for functions which do not ful�l the
assumption (66) can often be shown to be equivalent to a variational
problem for a that ful�ls (66). Assume that
~
is a homogeneous
free energy that does not necessarily ful�l (66). Let A be any a�ne
function which graph de�nes a supporting hyperplane for
~
in exactly
M � 2 points. This means
~
(c
0
) � A(c
0
) for all c
0
2 �
and
(c
0
) = A(c
0
) in exactly M points
(see Figure 4 for N = M = 2). Then we can subtract A from
~
to
obtain a function that ful�ls (66). Due to the fact that we impose an
integral constraint, the minimisation problem (P
"
) remains unchanged
by this procedure. In particular, all homogeneous free energies that
appear in the theory of phase separation can be reduced to ful�l (66)
in this way.
Under the assumption (66) it turns out that it is energetically favoura-
ble for c to attain the values p
1
; :::;p
M
. And in fact minimisers of E
"
in most of the domain have values close to p
1
; :::;p
M
. Our goal is
to show that under some natural growth assumptions on minimisers
of E
"
converge to solutions of a partition problem. To formulate this
partition problem we need to introduce some notation. We consider
partitions of into measurable sets
1
,...,
M
. These sets are assumed
to ful�l
M
X
k=1
X
k
= 1 almost everywhere in :
This means that up to a set of measure zero the sets
1
,...,
M
are a
partition of . We want to measure the area of the interface between
47
A
c
~ψ
Figure 4
two sets
k
and
l
, i.e. we want to measure the area of @
k
\ @
l
\
in an appropriate generalised sense. Here, it is convenient to use the
setting of functions of bounded variation. As general references to
functions with bounded variation we refer to the books of Evans and
Gariepy [35] and Giusti [47].
Assuming that the sets
1
; :::;
M
are sets with �nite perimeter in ,
i.e. X
1
; :::;X
M
lie in BV (), we can de�ne the interfacial measures
as
�
kl
=
1
2
(jrX
k
j+ jrX
l
j � jr(X
k
+ X
l
)j) :
This is a measure theoretic way to de�ne the (n�1){dimensional mea-
sure of the interface between the sets
k
and
l
. Introducing the
reduced boundary @
�
k
of the sets
k
it can be shown (see Bronsard,
Garcke and Stoth [16]) that for all open sets D �
�
kl
(D) = H
n�1
(@
�
k
\ @
�
l
\D) :
For D = and for sets
k
and
l
with su�ciently smooth boundaries
this shows that in fact �
kl
() measures the total interfacial area be-
tween
k
and
l
. Let us de�ne the interfaces between the sets
k
and
l
as
�
kl
= @
�
k
\ @
�
l
\ :
It will turn out that the �{limit E
0
of the functionals fE
"
g
">0
con-
sists of two summands. One measures the interfacial energy of the
individual interfaces �
kl
and the other takes elastic energy contribu-
tions into account. The interfacial energy is given as the sum of the
48
area of the interfaces �
kl
weighted by the individual surface tensions
�
kl
. These surface tensions are de�ned by means of the homogeneous
free energy . To make this precise we need to introduce the metric d
on � induced by
p
. For c
0
1
; c
0
2
2 � we de�ne
d(c
0
1
; c
0
2
) := inf
�
2
Z
1
�1
p
( (t))j
0
(t)j dt j : [�1; 1]! � (67)
is Lipschitz continuous, (�1) = c
0
1
and (1) = c
0
2
:
A curve that realizes the in�mum in the above expression is a geo-
desic with respect to this metric. The curve then realizes an inter-
facial layer with minimal energy
R
1
�1
fj
0
(t)j
2
+( (t))g dt. In fact,
Sternberg [89] proves that a geodesic connecting the points p
k
and p
l
when suitably reparametrised also solves the minimum problem
inf
�
2
Z
1
�1
�
j
0
(t)j
2
+( (t))
dt j : (�1;1)! �
is Lipschitz continuous, (�1) = p
k
and (1) = p
l
which is the variational problem of �nding an interfacial layer with
minimal energy. Stretching the t{variable by a factor
1
"
then gives
that the minimiser realises an interfacial layer which minimises the
�rst two terms in the energy E
"
when compared to all other one{
dimensional interfacial layers. This shows the importance of the metric
d and its geodesics when one tries to minimise the functional E
"
. Now
the surface tensions �
kl
are given as the distance in the metric d between
the minimisers p
k
and p
l
of , i.e.
�
kl
= d(p
k
;p
l
):
Our goal is to show that minimisers of E
"
converge (along subse-
quences) to minimisers of the functional
E
0
: L
1
(;�)�X
2
! IR [ f1g
with
E
0
(c;u) =
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
M
X
k;l=1
k<l
�
kl
H
n�1
(@
�
fc = p
k
g \ @
�
fc = p
l
g) +
Z
W (c; E(u))
if c 2 BV (); (c) = 0 a.e. ; �
Z
c =m;
1 otherwise:
Here, fc = p
k
g := fx 2 j c(x) = p
k
g. We note that c 2 BV ()
implies that the sets fc = p
k
g are sets of �nite perimeter in . This
49
follows from the co{area formula (see Appendix) applied to the function
x 7! d(c(x);p
k
) and will become clear from the discussion below.
Finding a minimiser of E
0
can be interpreted as a partition problem
for the set . De�ning for all E
0
2 IR
n�n
W
k
(E
0
) := W (p
k
; E
0
)
as the elastic energy density of phase k, the partition problem is as
follows.
Find a partition = [
M
k=1
k
with
k
\
l
= ; and
M
X
k=1
p
k
j
k
j
jj
=m (68)
such that the energy
M
X
k;l=1
k<l
�
kl
H
n�1
(@
�
k
\ @
�
l
) +
M
X
k=1
Z
k
W
k
(E(u))
becomes minimal. In the case that the convex hull of the vectors
p
1
; :::;p
M
is (M � 1){dimensional the constraint (68) �xes the volume
of the individual phases.
As has been used already, the constraints (c) = 0 a.e. and �
R
c =
m in the de�nition for E
0
imply (68). This means that the vector of
mean values is a convex combination of the minimal points p
k
. There-
fore we will require for the rest of the section that
m 2 convfp
1
; :::;p
M
g;
where convfp
1
; :::;p
M
g is the convex hull of the points p
1
; :::;p
M
. If
the mean value does not lie in the convex hull of p
1
; :::;p
M
it is an easy
matter to show that the limit of the minimal values of E
"
will be 1.
This means the sequence fE
"
g
">0
can only have the functional which
is identically equal to 1 as a �{limit.
Before proving a result on the limit of the functionals fE
"
g
">0
let us
state the results known on the �{limit of the functionals E
"
in the case
without elasticity. In the case N = 2 due to the constraint c
1
+ c
2
= 1
the problem can be reduced to a problem de�ned via a scalar quantity.
Modica [77] showed that if a sequence of minimisers c
"
to the variational
problem with W = 0 converges in L
1
() to a limit c, then the limit
de�nes a partition of with minimal interfacial area. His proof is
based on earlier work of him together with Mortola [79]. This result
was generalised by Fonseca and Tartar [38] and Sternberg [89] to the
vectorial case under the assumption that M = 2, i.e. only has two
global minimiser and hence only two phases are present. Both papers
50
also state assumptions on under which sequences (c
"
)
">0
with E
"
(c
"
)
uniformly bounded are compact in L
1
. Baldo [8] studied the vector{
valued problem with a �nite number of global minimisers of and
showed that L
1
{limits of minimisers of (P
"
) are minimisers of E
0
. We
also refer to work of Ambrosio [7], who studied the case in which the
set of zeros of can be any compact set in IR
N
. In particular, he
proved compactness in L
1
for sequences (c
"
)
">0
with bounded energy
E
"
(c
"
) under very general assumptions.
We make the following assumptions on .
(A3
00
) The homogeneous free energy 2 C
1
(IR
N
; IR) ful�ls (66) and
there exist constants c
4
; C
4
> 0 such that
(c
0
) � c
4
jc
0
j
2
� C
4
for all c
0
2 � :
The growth condition on will ensure compactness in L
2
for fc
"
g
">0
if the sequence f(c
"
;u
"
)g
">0
has uniformly bounded energy E
"
(c
"
;u
"
).
The compactness of c
"
will be necessary to handle the elastic part of
the free energy when we prove the following two theorems.
In the �rst theorem we state in part a) a compactness result for
sequences f(c
"
;u
"
)g
">0
with bounded energy E
"
(c
"
;u
"
) and in part b)
and c) the �{convergence of E
"
to E
0
is shown.
Theorem 5.1. Assume (A1), (A3
00
), (A4) andm 2 convfp
1
; :::;p
M
g.
a) Let f(c
"
;u
"
)g
">0
� H
1
(;�) \X
2
be such that �
R
c
"
=m and
E
"
(c
"
;u
"
) is uniformly bounded.
Then there exists a sequence f"
�
g
�2IN
� IR
+
with lim
�!1
"
�
= 0 and
c 2 L
2
(;�) \BV (;�), u 2 X
2
such that
c
"
�
! c in L
2
(;�);
u
"
�
! u weakly in H
1
(; IR
n
)
as "
�
tends to zero. Furthermore, it holds that c 2 fp
1
; :::;p
M
g almost
everywhere.
b) For all f(c
"
�
;u
"
�
)g
�2IN
2 H
1
(;�)\X
2
with c
"
�
! c in L
1
(;�)
and u
"
�
! u in L
2
(; IR
N
) as "
�
tends to zero, it holds
E
0
(c;u) � lim inf
�!1
E
"
�
(c
"
�
;u
"
�
):
c) For any (c;u) 2 L
1
(;�) \ X
2
and any sequence "
�
& 0, � 2 IN,
there exists a sequence f(c
"
�
;u
"
�
)g
�2IN
2 H
1
(;�) \ X
2
with c
"
�
!
c in L
1
(;�) and u
"
�
! u in L
2
(; IR
N
) as "
�
& 0 such that
E
0
(c;u) � lim sup
�!1
E
"
�
(c
"
�
;u
"
�
):
51
The preceeding theorem enables us to show that minimisers of E
"
converge (along subsequences) to minimisers of E
0
.
Theorem 5.2. Under the assumptions of Theorem 5.1 the variational
problems (P
"
) possess minimisers (c
"
;u
"
) 2 H
1
(;�) � X
2
provided
that " is small enough. Furthermore, there exists a sequence f"
�
g
�2IN
�
IR
+
with lim
�!1
"
�
= 0 and a (c;u) 2 L
2
(;�)�H
1
(; IR
n
) such that
i)
c
"
�
! c in L
2
(;�);
u
"
�
! u strongly in H
1
(; IR
n
);
ii) (c;u) is a global minimiser of E
0
. In particular
c 2 BV (; fp
1
; :::;p
M
g) and �
R
c =m.
Remark 5.1. We remark that we are able to prove strong convergence
of u
"
�
in H
1
(; IR
n
). This will be important later when we want to pass
to the limit in the Euler{Lagrange equation (see Section 6).
5.1. The �{limit of the elastic Ginzburg{Landau energies.
Proof of Theorem 5.1 a).
Since
W (c
0
; E
0
) � c
3
jE
0
j
2
� C
3
�
jc
0
j
2
+ 1
�
we can use the boundedness of E
"
(c
"
;u
"
) and the quadratic growth of
to conclude that
Z
�
"jrc
"
j
2
+
1
"
(c
"
)
�
(69)
is uniformly bounded if " is small enough. Our �rst goal is to show
compactness of (c
"
)
">0
. Here we use an idea of Ambrosio [7]. We de�ne
for all c
0
2 �
�
k
(c
0
) = d(c
0
;p
k
)
which is locally Lipschitz continuous with
jD
c
0
�
k
(c
0
)j � 2
p
(c
0
) a:e: in �:
For a proof see e.g. Fonseca and Tartar [38] and Ambrosio [7]. De�ning
for k = 1; :::;M and " > 0
f
"
k
(x) = min(�
k
(c
"
(x)) ; 1) ;
one can show via an approximation argument
jrf
"
k
(x)j � jD
c
0
�
k
(c
"
(x))j jrc
"
(x)j
� 2
p
(c
"
(x)) jrc
"
(x)j
� "jrc
"
(x)j
2
+
1
"
(c
"
(x))
52
for almost all x 2 . Using (69) it follows
f
"
k
is uniformly bounded in BV ()
for all k 2 f1; :::;Mg.
Since the embedding of BV () into L
1
() is compact we can con-
clude that there exists a subsequence f"
�
g
�2IN
� IR
+
tending to zero
such that for all k 2 f1; :::;Mg
f
"
�
k
! f
k
in L
1
() and a.e. as "
�
tends to 0:
The uniform boundedness of the term in (69) implies that
(c
"
�
)! 0 in L
1
() as "
�
tends to 0:
There exists a subsequence (which we again denote by f"
�
g
�2IN
) with
(c
"
�
)! 0 almost everywhere as "
�
! 0: (70)
De�ning the increasing function
!(z) = inff(c
0
) jdist(c
0
; fp
1
; :::;p
M
g) � z g
we have, due to the assumptions on (see (66) and (A3
00
)), that
!(z) > 0 if z > 0;
!(z)! 0 as z! 0:
Since
(c
0
) � ! (dist (c
0
; fp
1
; :::;p
M
g))
we get with (70)
dist
�
c
"
�
; fp
1
; :::;p
M
g
�
! 0 as "! 0 (71)
on nN , where N is a set with Lebesgue measure zero. Now we de�ne
k
= fx 2 n N j lim
�!1
d(c
"
�
(x);p
k
) = 0 g:
Claim 1:
lim
�!1
c
"
�
(x) = p
k
on
k
:
Suppose the claim were false. Then, due to the fact that
dist
�
c
"
�
; fp
1
; :::;p
M
g
�
! 0 as "
�
! 0;
we can �nd a x 2
k
, an l 2 f1; :::;Mg n fkg and a subsequence
f"
�
g
�2IN
� f"
�
g
�2IN
such that
c
"
�
(x)! p
l
as "
�
tends to zero:
Consequently
lim
"
�
!0
d(c
"
�
(x);p
k
) = d(p
l
;p
k
) 6= 0;
which is a contradiction and Claim 1 is proved.
53
Claim 2:
L
n
�
n
�
[
M
k=1
k
��
= 0:
There exists a set S � of measure zero, such that for all k 2
f1; :::;Mg f
"
�
k
converges pointwise on n S and such that (71) holds
on n S. For each point x 2 n S one can �nd an l 2 f1; :::;Mg and
a subsequence f"
�
g
�2IN
� f"
�
g
�2IN
such that c
"
�
(x) ! p
l
as "
�
tends
to zero. Hence,
f
"
�
l
(x)! 0 as "
�
tends to 0:
Since the whole sequence f
"
�
l
converges we obtain x 2
l
and Claim 2
is shown.
Hence we obtain that c
"
�
converges almost everywhere to a measurable
function c which only attains the values fp
1
; :::;p
M
g. Assumption
(A3
00
) implies
jc
"
�
j
2
�
1
c
4
�
(c
"
�
) + C
4
�
and since (c
"
�
) ! 0 in L
1
() we can deduce with the help of the
generalised Lebesgue theorem that
c
"
�
! c in L
2
():
Now we can use the uniform bound on E
"
(c
"
;u
"
), the assumptions
(A3
00
) and (A4) on and W respectively and Korn's inequality to
conclude
u
"
�
is uniformly bounded in H
1
(; IR
n
):
This implies the weak convergence of a subsequence and hence a) is
shown.
Proof of Theorem 5.1 b).
In case that lim inf
�!1
E
"
�
(c
"
�
;u
"
�
) =1 the conclusion holds and we
can assume without loss of generality that
lim
�!1
E
"
�
(c
"
�
;u
"
�
) <1 and c
"
�
! c a.e. :
Using assumptions (A3
00
), (A4) and Korn's inequality we obtain the
existence of a constant C such that
Z
�
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
+ ku
"
�
k
H
1� C <1: (72)
Since lim
�!1
R
(c
"
�
) = 0 we can conclude
(c
"
�
)! 0 a.e. (73)
54
and hence c 2 fp
1
; :::p
M
g almost everywhere. Let
R = 1 +maxfd(p
k
;p
l
) j k; l 2 f1; :::;Mgg
and de�ne
'
k
(c
0
) = min(d(c
0
;p
k
);R) :
Similar as in the proof of part a) of the theorem we can show that
Z
jr
�
'
k
(c
"
�
)
�
j �
Z
�
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
and therefore the left hand side is uniformly bounded for k = 1; :::;M .
The convergence of '
k
(c
"
�
) in L
1
() yields
Z
jr ('
k
(c)) j � lim inf
�!1
Z
jr
�
'
k
(c
"
�
)
�
j:
Taking the measure theoretic supremum
W
M
k=1
(see De�nition 8.4) of
the sequence of measures jr
�
'
k
(c
"
�
)
�
j then implies
M
_
k=1
jr ('
k
(c)) j() � lim inf
�!1
M
_
k=1
jr
�
'
k
(c
"
�
)
�
j() (74)
� lim inf
�!1
Z
�
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
(see also Remark 8.4). Then, using the identity
'
k
(c) =
M
X
l=1
d(p
k
;p
l
)X
fc=p
l
g
and applying the co{area formula gives
jr ('
k
(c)) j() =
Z
jr
�
P
M
l=1
d(p
k
;p
l
)X
fc=p
l
g
�
j
=
Z
1
�1
Z
jrX
f
(
P
M
l=1
d(p
k
;p
l
)X
fc=p
l
g
)
<�g
j d�
�
Z
�
k
0
Z
jrX
f
(
P
M
l=1
d(p
k
;p
l
)X
fc=p
l
g
)
<�g
j d�
= �
k
Z
jrX
fc=p
k
g
j;
where �
k
:= minfd(p
k
;p
l
) j l = 1; :::;M ; l 6= kg. This implies that
k
:= fc = p
k
g is a set of �nite perimeter in . In particular, the
reduced boundary @
�
fc = p
k
g and the interfaces �
kl
= @
�
k
\@
�
l
\
are de�ned.
55
Claim: For all open sets D � it holds
M
_
k=1
jr ('
k
(c)) j(D) =
M
X
l;m=1
l<m
d(p
l
;p
m
)H
n�1
(�
lm
\D) : (75)
The function '
k
(c) jumps on @
�
fc = p
l
g\@
�
fc = p
m
g by an amount
of jd(p
k
;p
l
)� d(p
k
;p
m
)j. Hence, with the help of the co{area formula
we can show for all open D � (see [92, 8] for details)
jr'
k
(c)j(D) =
M
X
l;m=1
l<m
jd(p
k
;p
l
)� d(p
k
;p
m
)jH
n�1
(�
lm
\D) : (76)
Taking the measure theoretic supremum in (76) we obtain
M
_
k=1
jr'
k
(c)j(D) (77)
=
M
X
l;m=1
l<m
max
k=1;:::;M
jd(p
k
;p
l
)� d(p
k
;p
m
)jH
n�1
(�
lm
\D) :
The claim now follows since a triangle inequality holds for d.
Inequality (74) and the representation (75) give that
M
X
l;m=1
l<m
d(p
l
;p
m
)H
n�1
(@
�
fc = p
l
g \ @
�
fc = p
m
g \ ) (78)
� lim inf
�!1
Z
�
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
:
It remains to show that the term
R
W (c
"
�
;u
"
�
) is lower semi-continuous.
We can use the growth condition (A3
00
) and the facts (72) and (73) to
deduce that
c
"
�
! c in L
2
(;�) and u
"
�
! u weakly in H
1
(; IR
n
):
The convexity of W (c
0
; E
0
) in the variable E
0
implies
Z
W (c
"
�
; E(u
"
�
))�W (c; E(u)) =
=
Z
W (c
"
�
; E(u
"
�
))�W (c
"
�
; E(u)) +
Z
W (c
"
�
; E(u)) �W (c; E(u))
�
Z
W
;E
(c
"
�
; E(u)) :
�
E(u
"
�
)� E(u)
�
+
Z
W (c
"
�
; E(u)) �W (c; E(u)):
56
Now the weak convergence of E(u
"
�
) in L
2
(; IR
n�n
), the strong con-
vergence of c
"
�
in L
2
(;�) and the growth conditions in (A4) give
lim inf
�!1
�
Z
W (c
"
�
; E(u
"
�
))�W (c; E(u))
�
� 0:
Together with (78) this yields the conclusion of part b) of the theorem.
Proof of Theorem 5.1 c).
In the case that no elastic e�ects are present, a proof of c) was given by
Baldo [8] who used ideas of Modica and Mortola [79, 77]. We generalise
the proof to the case that elastic terms are included. Assume (c;u) 2
H
1
(;�) \X
2
with
E
0
(c;u) <1
is given. Then there exist sets
1
; :::;
M
� such that
c =
M
X
k=1
p
k
X
k
a:e:
with X
k
2 BV () and
P
M
k=1
X
k
= 1. In this case Baldo [8] showed
for all sequences "
�
& 0 the existence of functions c
"
�
2 H
1
(;�) such
that
c
"
�
! c in L
1
(;�)
and
M
X
l;m=1
l<m
d(p
l
;p
m
)H
n�1
(@
�
fc = p
l
g \ @
�
fc = p
m
g)
� lim sup
�!1
Z
�
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
:
The idea of Baldo's proof is as follows. First he approximates the
sets
1
; :::;
M
by partitions consisting of polygonal domains. Then
the sharp jumps, separating sets
k
and
l
, are replaced by smooth
transition layers of a thickness proportional to ". The smooth transition
layers are taken to be appropriately scaled geodesics connecting the
values p
k
and p
l
. This assumes that a geodesic realizing the in�mum in
the de�nition of d(p
k
;p
l
) (see (67)) exists. If a minimiser does not exist
a curve connecting p
k
and p
l
whose energy 2
R
1
�1
p
( (t))j
0
(t)j dt
is close to d(p
k
;p
l
) has to be chosen instead of a geodesic.
This construction may violate the mass constraint. Since the error
in the mass constraint is only of order " it is possible to make a small
and su�ciently smooth perturbation of the function in the bulk of one
of the sets
k
, so that the mass constraint is met. Such a perturbation
57
only changes the energy to an amount that vanishes in the limit "
�
& 0.
For the details of the proof we refer to Baldo [8].
Since the sequence c
"
�
is constructed such that
Z
�
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
is bounded,
we can conclude as in the proof of part a) that
c
"
�
! c in L
2
(;�):
Then, the growth condition jW (c
0
; E
0
)j � C
2
(jE
0
j
2
+ jc
0
j
2
+ 1) and the
generalised convergence theorem of Lebesgue imply
Z
W (c
"
�
;u)!
Z
W (c;u) as �!1:
Hence, we can choose (c
"
�
;u
"
�
) := (c
"
�
;u) to obtain
E
0
(c;u) � lim sup
�!1
E
"
(c
"
�
;u
"
�
);
which shows part c) of the theorem. ut
Now we show that minimisers of E
"
converge (along subsequences)
to minimisers of E
0
.
Proof of Theorem 5.2.
The existence of global minimisers (c
"
;u
"
) of (P
"
) for " small can be
shown by the direct method taking into account the assumptions (A3
00
),
(A4) and Korn's inequality. We refer to Lemma 3.1 where the direct
method was used in a similar setting.
Since m 2 convfp
1
; :::;p
M
g one can �nd a partition of into sets
1
; :::;
M
such that X
1
; :::;X
M
2 BV () and such that the mass con-
straint (68) is ful�lled. This means that by de�ning d =
P
M
k=1
p
k
X
k
we obtain that the integral constraint �
R
d = m holds. By choosing
any v 2 X
2
we get a pair (d;v) such that E
0
(d;v) remains bounded.
Part b) of Theorem 5.1 now implies that E
"
(c
"
;u
"
) remains bounded
as " tends to zero. Hence by part a) of the previous theorem it follows
that there exists a sequence f"
�
g
�2IN
� IR
+
with lim
�!1
"
�
= 0 and
c 2 L
2
(;�), u 2 X
2
such that
c
"
�
! c in L
2
(;�);
u
"
�
! u weakly in H
1
(; IR
n
)
as "
�
tends to zero. In addition it holds that c 2 fp
1
; :::;p
M
g almost ev-
erywhere. Now by standard arguments in the theory of �{convergence
it follows that (c;u) is a minimiser of E
0
. For the convenience of the
58
reader we give the arguments in detail. Assume there were an element
(^c;^u) such that
E
0
(^c;^u) < E
0
(c;u):
Then we could �nd by part b) of Theorem 5.1 a sequence f(^c
"
�
;^u
"
�
)g
�2IN
2
H
1
(;�)\X
2
, with^c
"
�
!^c in L
1
(;�) and^u
"
�
!^u in L
2
(; IR
N
)
as "
�
! 0, such that
E
0
(^c;^u) � lim sup
�!1
E
"
�
(^c
"
�
;^u
"
�
):
This would imply
lim sup
�!1
E
"
�
(^c
"
�
;^u
"
�
) � E
0
(^c;^u)
< E
0
(c;u)
� lim inf
�!1
E
"
�
(c
"
�
;u
"
�
)
� lim inf
�!1
E
"
�
(^c
"
�
;^u
"
�
)
which is a contradiction.
It remains to show the strong convergence of u
"
�
in H
1
(; IR
n
). To
show this we use the fact that u
"
�
minimises^u 7! E
"
�
(c
"
�
;^u) in the
class X
2
. Hence, variations in the direction � 2 X
2
give
Z
W
;E
(c
"
�
; E(u
"
�
))r� = 0:
By choosing � = u
"
�
� u and using the strict monotonicity of W with
respect to the variable E
0
we obtain:
c
1
Z
jE(u
"
�
� u)j
2
�
Z
�
W
;E
(c
"
�
; E(u
"
�
))�W
;E
(c
"
�
; E(u))
�
:
�
E(u
"
�
� u)
�
= �
Z
�
W
;E
(c
"
�
; E(u))
�
:
�
E(u
"
�
� u)
�
:
Since u
"
�
! u weakly in H
1
(; IR
n
) and c
"
�
! c strongly in L
2
(;�)
as � ! 1 we can use the assumption (A4) and the inequality above
to obtain:
Z
jE(u
"
�
� u)j
2
! 0 as �!1:
Now Korn's inequality implies strong convergence of u
"
�
in H
1
(; IR
n
)
which proves the theorem. ut
59
5.2. Euler{Lagrange equation for the sharp interface func-
tional. In this subsection we compute the Euler{Lagrange equation
for a minimiser of E
0
, i.e. for a minimiser of the elastic partition
problem. In the following lemma we derive equations by varying the
independent variable in such a way that the volume constraint is met
for the variations. This way we obtain an identity for divergence free
variations.
Lemma 5.1. (Weak formulation of the Euler{Lagrange equa-
tion) Assume is a bounded domain with C
1
{boundary. Let (c;u) 2
BV ()�X
2
with �
R
c =m be a minimiser of E
0
.
Then
Z
M
X
k;l=1
k<l
�
kl
(r � � � �
k
� r��
k
)�
kl
(79)
+
M
X
k=1
Z
k
�
W
k
Id� (ru)
t
W
k;E
�
: r� = 0
for all � 2 C
1
(; IR
n
) with r � � = 0 and � � n = 0 on @. Here and
below we use the notation �
k
= �
rX
k
jrX
k
j
.
Remark 5.2. In Lemma 5.1 we only consider divergence free vector
�elds �. Hence, the identity (79) can be rewritten as
�
Z
M
X
k;l=1
k<l
�
kl
�
k
� r��
k
�
kl
�
M
X
k=1
Z
k
�
(ru)
t
W
k;E
�
: r� = 0: (80)
We stated the Euler{Lagrange equations in the above form because we
will consider variations along more general vector �elds � later.
Proof of Lemma 5.1. We consider the family of di�eomorphisms
�(�; �), � 2 [��
0
; �
0
] of given by
�(0;x) = x and �
;�
(�;x) = �(�(�;x))
for x 2 and � 2 [��
0
; �
0
]. The mappings �(�; �) de�ne a one para-
metric group of di�eomorphisms and in particular it holds
�(�;�(��;x)) = x;
i.e. �(�; �) is the inverse of �(��; �). Via the di�eomorphisms �(�; �)
we de�ne variations of the independent variable and obtain
c
�
(x) := c (�(��;x)) ;
u
�
(x) := u (�(��;x)) ;
�
k
(x) := �(�;
k
) = f�(�;x) jx 2
k
g:
60
Since
d
d�
det�
;x
(�;x) = (r � �) (�(�;x)) det�
;x
(�;x)
we get
d
d�
j
�
k
j =
d
d�
Z
k
jdet�
;x
(�;x)jdx
=
Z
k
(r � �) (�(�;x)) det�
;x
(�;x) dx
and since � is divergence free, deformations of by �(�; �) do not
change the volume of the individual phases. Hence the integral con-
straint
�
Z
c
�
=m
is ful�lled. Notice that (c;u) is also a minimiser if we allow u to vary in
the larger class H
1
(; IR
N
). Hence, (c
�
;u
�
) is allowed as a comparison
function and we obtain E
0
(c;u) � E
0
(c
�
;u
�
), which implies
0 =
d
d�
E
0
(c
�
;u
�
)
j�=0
if the derivative exists. Let us compute the above derivative. Using
the identity
�
kl
=
1
2
(jrX
k
j+ jrX
l
j � jr(X
k
+ X
l
)j)
one can reduce the computation of the derivative of the �rst term in E
0
to the problem of computing the �rst variation of area. Determining
the �rst variation of area in the setting of sets of bounded perimeter
is standard (see for example Giusti [47]) and we only sketch the argu-
ments. Let X
�
be either X
�
k
or X
�
k
[
�
l
= X
�
k
+ X
�
l
and X = X
0
.
Going back to the de�nition of the variation
R
jrX
�
j we obtain with
the help of the change of variable formula and an approximation argu-
ment
Z
jrX
�
j =
Z
j (�
;x
(�; �))
�1
�j jdet �
;x
(�; �)j jrX j; (81)
where � = �
rX
jrXj
is the generalised unit normal which is a jrX j{
measurable function. The right hand side is di�erentiable with respect
to the parameter � . Since
d
d�
(jdet�
;x
(�;x)j)
j�=0
= r � �(x) (82)
61
and
d
d�
�
(�
;x
(�;x))
�1
�
j�=0
= �r�(x) (83)
we obtain
d
d�
�
Z
jrX
�
j
�
j�=0
=
Z
(r � � � � � r��) jrX j:
De�ning �
k
= �
rX
k
jrX
k
j
it holds
�
k
+ �
l
= �
rX
k
jrX
k
j
�
rX
l
jrX
l
j
= 0 �
kl
{almost everywhere
and
r(X
k
+ X
l
)
jr(X
k
+ X
l
)j
= 0 �
kl
{almost everywhere
(see Theorem 8.7). Then we obtain using the representation of jrX
k
j
and jr(X
k
+ X
l
)j of Theorem 8.7
d
d�
�
H
n�1
(@
�
fc
�
= p
k
g \ @
�
fc
�
= p
l
g)
�
j�=0
=
d
d�
(�
�
kl
())
j�=0
=
d
d�
�
1
2
�
jrX
�
k
j+ jrX
�
l
j � jr(X
�
k
+ X
�
l
)j
�
()
�
j�=0
=
Z
(r � � � �
k
� r��
k
)�
kl
which gives the derivative of the �rst part of E
0
. It remains to compute
the derivative of
R
W (c
�
; E(u
�
)). It holds
Z
W (c
�
(y); E(u
�
)(y))dy =
Z
W (c(�(��;y)); E(u(�(��;y))) dy:
Setting x = �(��;y) and using �
;x
(�;�(��;y))�
;x
(��;y) = Id we
compute
E(u(�(��;y))) =
1
2
�
r
y
(u(�(��;y))) + (r
y
(u(�(��;y))))
t
�
=
1
2
�
ru(x)(�
;x
(�;x))
�1
+
�
ru(x)(�
;x
(�;x))
�1
�
t
�
:
62
Hence,
Z
W (c
�
(y); E(u
�
)(y))dy =
Z
W
�
c;
1
2
�
ru(�
;x
(�; �))
�1
+
�
ru(�
;x
(�; �))
�1
�
t
�
�
jdet�
;x
(�; �)j
where the integration in the last line was with respect to the variable x.
Using (82), (83) and the growth conditions on W and W
;E
(see (A4))
we compute
d
d�
�
Z
W (c
�
; E(u
�
))
�
j�=0
=
=
Z
�
W (c; E(u))r � � �W
;E
(c; E(u)) :
1
2
�
rur� + (rur�)
t
�
�
=
Z
h
W (c; E(u))r � � �W
;E
(c; E(u)) : (rur�)
i
=
Z
h
W (c; E(u))r � � �
�
(ru)
t
W
;E
(c; E(u))
�
: r�
i
where we used the symmetry of W
;E
to obtain the last two identities.
Since W (p
k
; �) = W
k
(�) the lemma is proved. ut
It is our goal to derive from the weak form of the Euler{Lagrange
equation conditions that hold locally in . We will derive conditions in
the bulk, on interfaces and on boundaries of interfaces. We sketch the
derivation of these conditions for absolute minimiser that are regular
enough.
We assume:
a) the sets
k
are Lipschitz,
b) the sets �
kl
= @
�
k
\ @
�
l
consist of a �nite number of oriented
C
2
{hypersurfaces where the orientation is given by �
k
. If @�
kl
6= ; we
assume that @�
kl
consists of a �nite number of C
1
{(n�2){dimensional
surfaces. It is furthermore assumed that these surfaces are either sub-
sets of @ or that they meet with the boundary of two or more other
interfaces.
c) u 2 H
2
(
k
; IR
n
),
d) W 2 C
2
.
Assumption c) implies thatru has traces on @
k
that lie in L
2
(@
�
k
; IR
n
).
We denote by div
�
kl
the tangential divergence with respect to the in-
terface �
kl
and by �
kl
the outer unit normal to @�
kl
. Then we compute
63
with the help of the Gau� theorem on manifolds
Z
(r � � � �
k
� r��
k
)�
kl
=
Z
�
kl
div
�
kl
� dH
n�1
=
Z
�
kl
(div
�
kl
�
k
) (� � �
k
) dH
n�1
+
Z
@�
kl
(� � �
kl
) dH
n�2
:
Since r �W
k;E
(E(u)) = 0 almost everywhere in
k
and since W
k;E
is
symmetric we obtain
r �
�
W
k
Id� (ru)
t
W
k;E
�
=
= rW
k
� (@
i
ru : W
k;E
)
i=1;:::;n
� (ru)
t
r �W
k;E
= 0
which holds almost everywhere in
k
. Hence,
Z
k
�
W
k
Id� (ru)
t
W
k;E
�
: r� = �
Z
k
r �
�
W
k
Id� (ru)
t
W
k;E
�
� �
+
Z
@
k
� �
�
W
k
Id� (ru)
t
W
k;E
�
�
k
=
Z
@
k
\
� �
�
W
k
Id� (ru)
t
W
k;E
�
�
k
where we also used that � �n = 0 and W
k;E
�n = 0 on @ to obtain the
last identity. Therefore, we obtain
M
X
k;l=1
k<l
�
kl
�
Z
�
kl
div
�
kl
�
k
(� � �
k
) dH
n�1
+
Z
@�
kl
� � �
kl
dH
n�2
�
(84)
+
M
X
k=1
Z
@
k
\
� �
�
W
k
Id� (ru)
t
W
k;E
�
�
k
dH
n�1
= 0
where �
kl
is the outward unit normal to @�
kl
. The above identity holds
for all � 2 C
1
(
�
) with r � � = 0 and � � n = 0 on @.
Now let � be a smooth (n � 2){dimensional subset of [
M
k;l=1
k<l
@�
kl
chosen in such a way that � is either a subset of @ or lies in the inter-
section of three or more interfaces. If � � @ we obtain by choosing
variations in the neighbourhood of a point in � that (84) leads to
�
kl
= n on @ \ @�
kl
: (85)
This says that the interface intersects the outer boundary orthogonal.
Note that we can only choose variations that lead to �'s with vanishing
normal component at the boundary of , which implies that �
kl
still
can have a normal component. Variations close to points on � have to
be done in such a way that the volume of the individual phases remains
64
�xed. If a variation close to a point on � changes volume, the volume
constraint still can be met by pushing an interface in the opposite di-
rection a bit further away. Taking into account that the integration
with respect to � is (n � 2){dimensional whereas the integration with
respect to the interfaces is (n� 1){dimensional we can obtain a point-
wise identity on � (i.e. the identity (85)) by choosing variations with
support closer and closer to �. We remark that in constructing these
variations we only need to ensure that the volume constraint is met. In
particular there is no need to ensure that volume is conserved locally,
which means that r � � = 0 in is not required necessarily.
If � lies in the interior of it has to be the junction of three or more
interfaces. We de�ne a set of tuples I � f1; :::;Mg
2
such that
� � @
�
�
kl
if and only if (k; l) 2 I:
By choosing variations locally around � we deduce that
X
(k;l)2I
�
kl
�
kl
= 0 on �: (86)
Here again we need to take care of the volume constraint. And again
we use the fact that the sets over which we integrate in (84) have
di�erent dimension to conclude (86). The identity (86) is the well
known Young's law which can be interpreted as a force balance at
multiple junctions [100, 48, 16, 41]. Young's law implies conditions for
the angles at which interfaces can intersect. For example in the case
that three interfaces, which are denoted by (1; 2); (2; 3); (1; 3), meet at
a triple junction we obtain
�
12
�
12
+ �
23
�
23
+ �
13
�
13
= 0;
which implies that the angles �
1
; �
2
; �
3
between the three surfaces
�
12
;�
23
;�
13
ful�l the condition
�
12
sin �
3
=
�
23
sin �
1
=
�
13
sin �
2
;
where we denote by �
1
the angle between �
12
and �
13
, by �
2
the angle
between �
12
and �
23
and by �
3
the angle between �
13
and �
23
. For more
discussion on Young's law see [100, 16, 41]. If all surface energies are
the same, we obtain that three interfaces meet at 120
�
angles, which
are the same as the angles at which surfaces making up soap bubble
clusters meet. Bubble clusters realize the least area way to enclose
and separate several regions of prescribed volumes (see e.g. Morgan
[80], Chapter 13). This would be a solution of the limiting variational
problem we considered above in the case that the elastic energy is zero
and that all surface tensions are the same.
65
We have shown that for an absolute minimiser the (n�2){dimensional
integral in (84) is zero and we obtain
T (�) :=
M
X
k;l=1
k<l
�
kl
Z
�
kl
div
�
kl
�
k
(� � �
k
) dH
n�1
(87)
+
M
X
k=1
Z
@
k
\
� �
�
W
k
Id� (ru)
t
W
k;E
�
�
k
dH
n�1
= 0
for � 2 C
1
() with r � � = 0 and � � n = 0 on @. The functional
T when de�ned by (87) is a bounded linear functional on H
1
(; IR
n
)
with T (�) = 0 for all divergence free � 2 H
1
0
(; IR
n
). This implies the
existence of a function p 2 L
2
() such that
T = rp
(see Temam [90]), i.e.
Z
p r � � =
M
X
k;l=1
k<l
�
kl
Z
�
kl
div
�
kl
�
k
(� � �
k
) dH
n�1
(88)
+
M
X
k=1
Z
@
k
\
� �
�
W
k
Id� (ru)
t
W
k;E
�
�
k
dH
n�1
for all � 2 H
1
0
(; IR
n
). Since the left hand side of (88) is zero if � is
supported in
k
, we obtain that p is constant on all connected subsets
of
k
. The sum of the principal curvatures (i.e. (n�1){times the mean
curvature) of the interface �
kl
is given by
�
kl
= div
�
kl
�
k
on �
kl
:
Now we choose a function � which ful�ls � = V �
k
on �
kl
where V is
an arbitrary su�ciently smooth function de�ned on �
kl
with compact
support in the interior of �
kl
. It is furthermore assumed that � is chosen
such that it is zero on all other interfaces (see for example [47], Chapter
10, for details on the construction of such functions �). Making use of
the fact that V is arbitrary gives
�
kl
�
kl
+ �
k
�
WId� (ru)
t
W
;E
�
k
l
�
k
= [p]
k
l
on �
kl
: (89)
For a quantity q that jumps across an interface �
kl
we de�ne
[q]
q
l
:= q
l
� q
k
;
66
where q
k
and q
l
are the traces of q on �
kl
with respect to
k
and
l
respectively. The term �
k
�
WId� (ru)
t
W
;E
�
k
l
�
k
is the Eshelby trac-
tion and p plays the role of a Lagrange multiplier taking into account
that the volume of the phases is prescribed.
A minimiser of the energy E
0
ful�ls
r �W
;E
(c;u) = 0
in the sense of distributions. This implies that the normal component
of W
;E
does not jump across the boundary of
k
. Hence, it holds
[W
;E
]
k
l
�
k
= 0 on �
kl
:
This implies that (89) can be rewritten as
�
kl
�
kl
+ [W ]
k
l
� [ru �
k
]
k
l
� (W
;E
�
k
) = [p]
k
l
on �
kl
(90)
whereW
;E
�
k
can be determined by taking the trace ofW
;E
on �
kl
either
with respect to
k
or with respect to
l
. The termW
;E
�
k
is the traction
at the interface (see [40]). The identity (90) is a stress modi�ed version
of the Gibbs{Thomson equation (see e.g. [20, 55, 66, 51, 40]).
Below we show that the function p cannot attain di�erent values
in parts of
k
that are not connected. To show this we still suppose
that (c;u) is an absolute minimiser of the functional E
0
. Hence we
can also allow for variations that change the volume of the connected
components, if one guarantees that the overall volume of the phases
remains the same. Again we consider a situation in which the minimiser
(c;u) ful�ls the regularity conditions a){d). Let us sketch the idea of
the argument which shows that p is constant on
k
.
Assume there are two maximally connected subsets
0
1
and
0
2
of
k
0
, k
0
2 f1; :::;Mg. Then there exists a chain of sets
0
1
= D
1
;D
2
; :::;D
m�1
;D
m
=
0
2
whose interiors are mutually disjoint and which are such that each of
the sets D
1
;D
2
; :::;D
m�1
;D
m
is a maximally connected subset of one
the sets
1
; :::;
M
and such that
H
n�1
(@
�
D
�
\ @
�
D
�+1
) > 0 for � = 1; :::;m� 1:
If such a chain would not exist, we could deduce that a partition of
into two sets with non-vanishing volume and zero interfacial area
exists. But this is a contradiction to the isoperimetric inequality for
partitions of .
The idea now is to push a certain amount of volume from
0
1
through
the sets D
2
; :::;D
m�1
into the set
0
2
which by assumption belongs to
the same phase as the set
0
1
. Then we want to exploit the fact that
this variation cannot increase the energy and hence the derivative of
67
the energy along this variation has to be zero. We choose families of
di�eomorphisms �
�
(�; �), � 2 [��
0
; �
0
], � 2 f1; :::;m� 1g, such that
�
�
(�; �) = Id on all interfaces besides @
�
D
�
\ @
�
D
�+1
;
�
�;�
(0;x) = V
�
(x)�
�
(x)
where V
�
is a su�ciently smooth function on @
�
D
�
\ @
�
D
�+1
with
compact support and �
�
is the unit normal on @
�
D
�
\@
�
D
�+1
pointing
from D
�
into D
�+1
. We choose the functions V
�
such that
Z
@
�
D
�
\@
�
D
�+1
V
�
dH
n�1
= 1:
Choosing di�eomorphisms �
�
(�; �) with normal velocities V
�
on D
�
\
@
�
D
�+1
and such that �
�
is constant on the other interfaces we obtain
that the volume of the sets D
�
(� ) changes in the following way
d
d�
jD
1
(� )j = 1;
d
d�
jD
�
(� )j = 0; � = 2; :::;m� 1;
d
d�
jD
m
(� )j = �1:
This can be done by an appropriate normalisation of the typical con-
struction of normal variations on an interface (see e.g. Giusti [47]).
The equations above guarantee that the composition � of the (m�1){
di�eomorphisms �
1
; :::;�
m�1
preserves the total volume of the individ-
ual phases. Computing the �rst variation
d
d�
E
0
(c
�
;u
�
)
j�=0
using the
smoothness assumptions from above yields
0 =
m�1
X
�=1
Z
@
�
D
�
\@
�
D
�+1
V
�
n
�
k
�
l
�
�
�
+ �
�
�
WId� (ru)
t
W
;E
�
�+1
�
�
�
o
dH
n�1
:
By �
�
we denote (n � 1) times the mean curvature of @
�
D
�
\ @
�
D
�+1
and �
k
�
l
�
is the surface tension related to @
�
D
�
\ @
�
D
�+1
, i.e. k
�
; l
�
2
f1; :::;Mg are chosen such that @
�
D
�
\ @
�
D
�+1
belongs to �
k
�
l
�
. The
term in the brackets f�g is equal to the jump of p across the interface
(see (89)). Denoting by p
�
the value of p on D
�
we obtain
0 =
m�1
X
�=1
(p
�+1
� p
�
)
Z
@
�
D
�
\@
�
D
�+1
V
�
=
m�1
X
�=1
(p
�+1
� p
�
)
= p
m
� p
1
which proves that p
�
is constant on each of the sets
1
; :::;
M
.
68
Let us summarise the conditions a regular absolute minimiser of E
0
has to ful�l.
Strong formulation of the Euler{Lagrange equations
Assume an absolute minimiser of E
0
ful�ls the regularity assumptions
a){d).
Then there exists a Lagrange multiplierp = (p
1
; :::; p
M
) with
P
M
k=1
p
k
=
0 such that
In the sets
k
it holds:
1.)
r � [W
k;E
(E(u))] = 0:
On the boundary of it holds:
2.)
W
k;E
n = 0:
On the interfaces �
kl
it holds
3.)
�
kl
�
kl
+ �
k
�
WId� (ru)
t
W
;E
�
k
l
�
k
= p
k
� p
l
;
4.)
[W
;E
�
k
]
k
l
= 0;
5.)
[u]
k
l
= 0:
For co-dimension{2 junctions � of three or more interfaces we de�ne a
set of tuples I � f1; :::;Mg
2
such that
� � @
�
�
kl
if and only if (k; l) 2 I:
6.) On these junctions Young's law has to hold, i.e.
X
(k;l)2I
�
kl
�
kl
= 0 on �:
7.) Interfaces that intersect the outer boundary do these with a right
angle.
Remark 5.3. 1.) Condition 5.) means that the interface is coherent;
i.e. two phases neither separate nor slip at the interface. This im-
plies that [ru]
k
l
� = 0 for all � that are tangent to �
kl
and hence, the
tangential part of the gradient does not jump. This together with (89)
conditions 3.) and 4.) gives
�
kl
�
kl
�
k
+
�
WId� (ru)
t
W
;E
�
k
l
�
k
= (p
k
� p
l
)�
k
on �
kl
;
69
2.) The argument in 1.) together with the discussion leading to the
strong formulation of the Euler{Lagrange equations shows that any
function ful�lling the regularity requirements a){d) and the conditions
1.){7.) above also ful�ls
Z
(r � �) p =
M
X
k;l=1
k<l
�
kl
(r � � � �
k
� r��
k
) �
kl
+
+
Z
k
�
W
k
Id� (ru)
t
W
k;E
�
: r�
for all for all � 2 C
1
(; IR
n
) with � � n = 0 on @. Here, p is given
by p � p
k
on
k
.
6. The Gibbs{Thomson equation as a singular limit in the
scalar case
In this section we study the asymptotic limit of the di�use inter-
face model in the binary case. Due to the constraint c
1
+ c
2
= 1 we
can express the dependence on the concentration vector by the scalar
quantity
c = c
1
� c
2
which is the di�erence of the two concentrations.
The free energy then has the form (after normalising constants)
E
"
(c;u) :=
Z
�
"jrcj
2
+
1
"
(c) +W (c; E(u))
�
; " > 0;
where : IR! IR is assumed to have two global minimiser with height
zero. For simplicity of the presentation we rescale such that attains
the global minimum zero at �1, i.e.
� 0 and (c
0
) = 0 , c
0
= �1: (91)
We want to show that we can pass to the limit in the Euler{Lagrange
equation for a minimiser of E
"
to obtain the modi�ed Gibbs{Thomson
equation studied in the previous section. The result we obtain gen-
eralises a result of Luckhaus and Modica [73] to the case that elastic
energy contributions are taken into account.
In the whole of this section we assume
(B1) is a bounded domain with C
1
{boundary,
(B2) there exist constants c
4
; C
4
> 0 such that
(c
0
) � c
4
jc
0
j
2
� C
4
for all c
0
2 IR:
70
Furthermore it is assumed that =
1
+
2
where
1
is non{negative
and convex and
2
;c
has sub-linear growth. In addition we assume that
the growth of
1
;c
is bounded by
1
in the sense that for all � > 0 there
exists a constant C
�
such that
j
1
;c
(c
0
)j � �
1
(c
0
) + C
�
:
(B3) the assumptions (A4) for the elastic energy density W hold.
Under these assumptions one can show analogously to the vector case
the existence of an absolute minimiser of the following scalar minimum
problem.
(P
"
) Find a minimiser of
E
"
(c;u) :=
Z
�
"jrcj
2
+
1
"
(c) +W (c; E(u))
�
; " > 0
in the class of all functions (c;u) 2 H
1
()�X
2
that ful�l the constraint
�
R
c = m, where m 2 (�1; 1) is a given constant.
In the previous section we showed that in the limit "& 0 these varia-
tional problems lead to
(P
0
) Find a minimiser of
E
0
(c;u) := �
Z
jrcj+
Z
W (c; E(u));
in the class of all functions (c;u) 2 BV () � X
2
that ful�l the con-
straints c 2 f�1; 1g a.e. and �
R
c = m, where m 2 (�1; 1) is a given
constant.
In the following theorem we state the �rst variation of problem (P
"
)
with respect to the independent variable. Our goal later is to pass to
the limit " & 0 in the �rst variation formula that we obtain. This
is appropriate since for problem (P
0
) variations with respect to the
dependent variable c are not possible. This is due to the fact that for
the limiting problem c is constrained to attain the values �1.
71
Theorem 6.1. Under the assumptions (B1){(B3) a pair (c
"
;u
"
) 2
H
1
() \X
2
that is a solution to the variational problem (P
"
) ful�ls
Z
(
�
"jrc
"
j
2
+
1
"
(c
"
)
�
r � � � 2"rc
"
� r�rc
"
+ (92)
+
�
W (c
"
; E(u
"
))Id� (ru)
t
W
;E
(c
"
; E(u
"
))
�
: r�
)
=
Z
�
"
c
"
r � �
for all � 2 C
1
(; IR
n
) with � � n = 0 on @. Here,
�
"
= �
Z
�
1
"
;c
(c
"
) +W
;c
(c
"
; E(u
"
))
�
(93)
is a Lagrange multiplier.
Proof. In the proof we omit the index ". Let � 2 C
1
(
�
; IR
n
) be
such that � � n = 0 on @. Then we choose a one parameter family
of di�eomorphisms �(�; �), � 2 [�
0
; �
0
] of de�ned via the solutions of
the following initial value problems
�(0;x) = x and �
;�
(�;x) = �(�(�;x))
where x 2 and � 2 [��
0
; �
0
]. We de�ne
c
�
(x) = c(�(��;x))��
Z
c(�(��;y))dy+m; (94)
u
�
(x) = u(�(�;x))
which is allowed as a comparison function in the minimisation problem
(P
"
). Note also that c
0
= c. Now we want to compute
d
d�
�
Z
�
"jrc
�
j
2
+
1
"
(c
�
) +W (c
�
; E(u
�
))
��
j�=0
:
This term is zero because c is the minimum in (P
"
). First we compute
Z
jrc
�
(y)j
2
dy =
Z
j (�
;x
(�;x))
�t
rc(x)j
2
jdet�
;x
(�;x)j dx
and obtain
d
d�
�
Z
jrc
�
j
2
�
j�=0
=
Z
�
jrcj
2
r � � � 2rc � r�rc
:
When we compute the derivative of the {term the mass correction
in (94) will give a term which is a summand in the formula for the
72
Lagrange multiplier. We get
d
d�
�
Z
(c
�
(y))dy
�
j�=0
=
d
d�
�
Z
�
c(x)��
Z
c(�(��;y))dy+m
�
jdet�
;x
(�;x)j dx
�
j�=0
=
Z
(c)r � � �
Z
�
;c
(c)
�
�
Z
cr � �
��
=
Z
(c)r � � �
Z
�
cr � �
�
�
Z
;c
(c)
��
:
For the elastic term we compute similar as above
Z
W (c
�
(y); E(u
�
(y))dy =
=
Z
W
�
c��
Z
c
�
+m;
1
2
�
ru(�
;x
)
�1
+
�
ru(�
;x
)
�1
�
t
�
�
jdet�
;x
j:
Hence,
d
d�
�
Z
W (c
�
(y); E(u
�
(y))dy
�
j�=0
=
=
Z
�
W r � � �
�
(ru)
t
W
;E
)
�
: r� �W
;c
�
�
Z
cr � �
��
where W , W
;E
and W
;c
in the last line are evaluated at (c; E(u)). This
completes the proof of the theorem. ut
Remark 6.1. Let us point out that the Lagrange multiplier �
"
also
ful�ls the identity
Z
�
"rc
"
r� +
1
"
;c
(c
"
)� +W
;c
(c
"
; E(u
"
))�
�
=
Z
�
"
� (95)
for all � 2 L
1
()\H
1
() which are the Euler{Lagrange equations one
obtains by variations with respect to the dependent variable (take � � 1
in (95) to obtain (93)). Luckhaus and Modica [73] who considered the
case without elasticity started with these equations and set � = c
"
r � �
and then derived the identity (92). Formally it is also possible to derive
equation (92) from (95) in the case with elastic interactions. We did
not follow this strategy because in the case with elasticity there is not
enough regularity known to make the formal calculations rigorous.
In the next theorem we state that the Lagrange multipliers �
"
con-
verge (along subsequences as " tends to zero) to a Lagrange multiplier
of the limiting partition problem (P
0
). This shows that the equation
73
for the chemical potential leads to the Gibbs Thomson law in the limit
as " tends to zero.
Theorem 6.2. Suppose (B1){(B3) are ful�lled and assume (c
"
;u
"
)
">0
are solutions of the variational problems (P
"
). Then for each subse-
quence ("
�
)
�2IN
& 0 such that
c
"
�
! c in L
1
(); (96)
u
"
�
! u in L
2
(; IR
n
) (97)
it holds
�
"
�
! �;
where � is a Lagrange multiplier for the minimum problem (P
0
), i.e.
Z
� (r � � � � � r��) jrX
fc=�1g
j +
Z
�
WId� (ru)
t
W
;E
�
: r�
=
Z
�cr � � (98)
for all � 2 C
1
(; IR
n
) with � � n = 0 on @. Here, � = �
rX
fc=�1g
jrX
fc=�1g
j
is
the generalised outer unit normal to fc = �1g which is a jrX
fc=�1g
j{
measurable function.
Remark 6.2. 1.) Theorem 5.1 c) yields the existence of a subsequence
such that the converge properties (96), (97) hold.
2.) Since E
0
is the �{limit of E
"
we conclude that (c;u) is an absolute
minimiser of E
0
.
3.) Notice that the term �c appearing in the term on the right hand
side of (98) is constant in the sets fc = �1g and fc = 1g. Setting
p = �c we obtain the formulation of the Euler{Lagrange equation used
in the previous section.
Proof of Theorem 6.2. As in the proof of Theorem 5.2 we can con-
clude that the convergence is even stronger than (96), (97). Precisely,
we obtain
c
"
�
! c in L
2
(;�);
u
"
�
! u in H
1
(; IR
n
);
E
"
�
(c
"
�
;u
"
�
) ! E
0
(c;u):
We also derive that (c;u) is a global minimiser of E
0
and
c 2 f�1; 1g almost everywhere.
74
The above convergence properties together with the growth condition
on W imply
Z
W (c
"
�
;u
"
�
)!
Z
W (c;u);
which yields that also
Z
�
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
! �
Z
jrX
fc=�1g
j (99)
where � = d(�1; 1) = 2
R
1
�1
p
(z) dz: Here we used the fact that the
geodesic realizing the in�mum in (67) is always realized by a straight
connection of two points on the real line. Hence, we compute
d(c
0
1
; c
0
2
) = 2
Z
c
0
2
c
0
1
p
(z) dz for all c
0
1
< c
0
2
:
Our goal is to pass to the limit in the equation
Z
��
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
r � � � 2"
�
rc
"
�
� r�rc
"
�
+ (100)
�
W (c
"
�
; E(u
"
�
))Id� (ru
"
�
)
t
W
;E
(c
"
�
; E(u
"
�
))
�
: r�
�
=
Z
�
"
�
c
"
�
r � �
to obtain the weak formulation of the Gibbs{Thomson equation. Here
it comes into play that we were able to show strong convergence of
ru
"
�
in L
2
(; IR
n
). Taking the growth condition on W and W
;E
into
account we can pass to the limit in the terms involving W and W
;E
to
obtain
Z
�
W (c
"
�
; E(u
"
�
))Id� (ru)
t
W
;E
(c
"
�
; E(u
"
�
))
�
: r�!
Z
�
W (c; E(u))Id� (ru)
t
W
;E
(c; E(u))
�
: r�
as "
�
tends to zero. It remains to pass to the limit in the terms involving
rc
"
�
and . To obtain the limit we can use the ideas of Luckhaus and
Modica [73] who studied the case without elastic energy contributions.
For completeness we present the details.
Similarly as in Section 5 we de�ne
�(c
0
) = d(c
0
;�1) = 2
Z
c
0
�1
p
(z) dz
and obtain
�(c
"
�
)!
(
� = �(�1) if z 2 fc = 1g,
0 if z 2 fc = �1g
(101)
75
almost everywhere. In addition, we have for almost all x
jr�(c
"
�
(x))j = �
;c
(c
"
�
(x))jrc
"
�
(x)j
= 2
p
(c
"
�
(x))jrc
"
�
(x)j (102)
�
1
"
�
(c
"
�
(x)) + "
�
jrc
"
�
(x)j
2
:
This implies that r�(c
"
�
) is uniformly bounded in L
1
(). A sequence
of functions converging almost everywhere and whose weak derivatives
are uniformly bounded in L
1
() also converges in L
1
(). Hence we
use (101) to conclude that �(c
"
�
) ! �(c) in L
1
(). Using the lower
semi-continuity property of BV {functions we conclude
Z
jr(�(c))j � lim inf
�!1
Z
2
p
(c
"
�
(x))jrc
"
�
(x)j
� lim inf
�!1
Z
�
1
"
�
(c
"
�
(x)) + "
�
jrc
"
�
(x)j
2
�
:
On the other hand with the same arguments as in the proofs of the
Theorems 5.1 c) and 5.2 we obtain (see (99))
Z
jr(�(c))j = lim
�!1
Z
�
1
"
�
(c
"
�
(x)) + "
�
jrc
"
�
(x)j
2
�
and
Z
jr(�(c))j = lim
�!1
Z
jr(�(c
"
�
(x))j (103)
= lim
�!1
Z
2
p
(c
"
�
(x))jrc
"
�
(x)j
= lim
�!1
Z
�
1
"
�
(c
"
�
(x)) + "
�
jrc
"
�
(x)j
2
�
:
Hence,
lim
�!1
Z
�
�
�
�
1
"
�
(c
"
�
) + "
�
jrc
"
�
j
2
� jr�(c
"
�
)j
�
�
�
�
=
lim
�!1
Z
�
1
"
�
(c
"
�
) + "
�
jrc
"
�
j
2
� 2
p
(c
"
�
)jrc
"
�
j
�
= 0:
Therefore,
lim
�!1
Z
�
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
r � � = lim
�!1
Z
r � �jr�(c
"
�
)j:
We now show that the limit on the right hand side is equal to
Z
r � �jr�(c)j :
76
In order to conclude this we need to show
jr�(c
"
�
)j ! jr�(c)j in the sense of Radon measures. (104)
This is a consequence of (103) and can be deduced as follows. For all
compact sets K � the lower semi-continuity of jr�(c
"
�
)j on open
sets implies
lim sup
�!1
Z
K
jr�(c
"
�
)j = lim sup
�!1
�
Z
jr�(c
"
�
)j �
Z
nK
jr�(c
"
�
)j
�
�
Z
jr�(c)j �
Z
nK
jr�(c)j =
Z
K
jr�(c)j:
Since a sequence of Radon measures converges weak-* if and only if
it is lower semi-continuous on open sets and upper semi-continuous
on compact sets (see Evans and Gariepy [35]) the convergence (104)
follows.
Using that
0 = lim
�!1
Z
�
1
"
�
(c
"
�
) + "
�
jrc
"
�
j
2
� 2
p
(c
"
�
)jrc
"
�
j
�
= lim
�!1
Z
r
1
"
�
(c
"
�
)�
p
"
�
jrc
"
�
j
!
2
(105)
and that
R
�
"
�
jrc
"
�
j
2
+
1
"
�
(c
"
�
)
�
is uniformly bounded, we obtain
lim
�!1
Z
�
�
�
�
1
"
�
(c
"
�
)� "
�
jrc
"
�
j
2
�
�
�
�
=
= lim
�!1
Z
�
�
�
�
�
r
1
"
�
(c
"
�
)�
p
"
�
jrc
"
�
j
!
r
1
"
�
(c
"
�
) +
p
"
�
jrc
"
�
j
!
�
�
�
�
�
= 0
which can be interpreted as equipartition of energy. This terminol-
ogy is chosen because the two summands making up the energy term
R
�
j"
�
rc
"
�
j
2
+
1
"
�
(c
"
�
)
�
are approximately equal for small "
�
. Iden-
tity (105) and the fact that
r�(c
"
�
) = �
;c
(c
"
�
)rc
"
�
= 2
p
(c
"
�
)rc
"
�
a.e.
77
gives
lim
�!1
Z
2"
�
rc
"
�
� r�rc
"
�
= lim
�!1
Z
2"
�
rc
"
�
jrc
"
�
j
� r�
rc
"
�
jrc
"
�
j
jrc
"
�
j
2
= lim
�!1
Z
rc
"
�
jrc
"
�
j
� r�
rc
"
�
jrc
"
�
j
p
(c
"
�
)jrc
"
�
j
= lim
�!1
Z
r�(c
"
�
)
r�(c
"
�
)j
� r�
r�(c
"
�
)
jr�(c
"
�
)j
jr�(c
"
�
)j:
It remains to pass to the limit in the last identity. De�ning the abbre-
viations �
"
�
:= �
r�(c
"
�
)
jr�(c
"
�
)j
and � := �
r�(c)
jr�(c)j
we need to show
lim
�!1
Z
�
"
�
� r��
"
�
jr�(c
"
�
)j =
Z
� � r��jr�(c)j: (106)
To conclude we construct a smooth approximation of the normals �
and show that these approximations are also good approximations for
�
"
�
(see also [74, 43, 16]). In fact since �(c) 2 BV () we obtain the
existence of approximative normals g
�
2 C
1
0
(; IR
n
) with jg
�
j � 1 such
that
Z
�
1 � g
�
� �
�
jr�(c)j � �:
Since
r�(c
"
�
)! r�(c) in the sense of measures
and since (103) holds, we obtain
lim
�!1
Z
(1 � g
�
� �
"
�
)jr�(c
"
�
)j = lim
�!1
Z
�
jr�(c
"
�
)j+ g
�
� r�(c
"
�
)
�
=
Z
jr�(c)j+
Z
g
�
� r�(c)
=
Z
�
1 � g
�
� �
�
jr�(c)j � �:
Using that g
�
and �
"
�
have norm less or equal to one we compute
�
�
�
"
�
� g
�
�
�
2
= j�
"
�
j
2
� 2 g
�
�
"
�
+ jg
�
j
2
� 2 (1 � g
�
� �
"
�
):
The last two computations give
lim
�!1
Z
�
�
�
"
�
� g
�
�
�
2
jr�(c
"
�
)j � �:
78
Therefore, we can conclude
lim sup
�!1
�
Z
�
"
�
� r��
"
�
jr�(c
"
�
)j �
Z
� � r��jr�(c)j
�
=
lim sup
�!1
�
Z
g
�
� r�g
�
jr�(c
"
�
)j � g
�
� r�g
�
jr�(c)j
�
+O(�)
which shows (106) because jr�(c
"
�
)j ! jr�(c)j in the sense of mea-
sures and since � can be chosen arbitrarily small.
It remains to prove convergence of the Lagrange multipliers �
"
�
. Here
we choose a � 2 C
1
0
(; IR
n
) with the property that
Z
r � � c > 0: (107)
This is possible since c 2 f�1; 1g almost everywhere and since �
R
c 2
(�1; 1) which implies that �
R
jrcj =
R
jr�(c)j 6= 0. Choosing such
a � in (100) we can conclude convergence of the Lagrange multipliers
from the convergence of the left hand side and the fact that
lim
"
�
!0
R
(r � �) c
"
�
=
R
(r � �) c > 0.
ut
7. Discussion
In this section we brie y state the main results obtained and we dis-
cuss some issues that are left for further studies. We tried to answer
some basic mathematical questions arising in the modelling of phase
separation in the presence of elastic interactions. Existence and unique-
ness theorems for Cahn{Hilliard systems with elasticity for smooth and
logarithmic free energies have been proved. Nevertheless, it is still not
clear whether solutions to these systems under general assumptions are
smooth. This is due to the fact that the elliptic system for the displace-
ment contains solution dependent coe�cients and it is well known that
results on everywhere regularity for elliptic systems in these cases are
very limited (see e.g. the book of Giaquinta [45]). For the case of
homogeneous elasticity it seems to be possible to show higher regular-
ity due to the constant coe�cients of the elliptic system. Then, one
can di�erentiate the equations to obtain higher regularity. This has to
be done by using a di�erence quotient method and will be done in a
further study.
We were able to show that the minimisers of the Ginzburg{Landau
free energy converge to minimisers of a sharp interface partition prob-
lem when the interfacial thickness tends to zero. Passing to the limit in
79
the corresponding Euler{Lagrange equation was possible in the case of
binary systems. We recovered an elastically modi�ed Gibbs{Thomson
law in the asymptotic limit. To generalise this result to more com-
ponents would require a generalisation of the result of Luckhaus and
Modica [73] to the vector valued situation which is not known yet.
Moreover, we studied convergence of minimisers of the Ginzburg{
Landau functional, i.e. convergence of globally stable stationary so-
lutions to the Cahn{Hilliard system. To identify the asymptotic limit
in the vector valued evolution problem with elasticity one would have
to combine the matched formal asymptotic expansions of Leo, Lowen-
grub and Jou [65] (who studied the binary case with elasticity) and
Bronsard, Garcke and Stoth [16] (who studied the multi{component
case without elasticity). It should be possible to use the methods of
Luckhaus and Sturzenhecker [68] and Bronsard, Garcke and Stoth [16]
to obtain a conditional existence result for the vector valued sharp
interface model with elastic contributions.
It is desirable to have a regularity theory for minimisers of the sharp
interface functional. In the two phase case the main question is whether
it is possible to show that minimisers are almost area minimising in the
sense of Almgren [1]. Then the general regularity theory for minimal
surfaces is applicable [37]. In this context we refer to work of Lin [69],
who developed a regularity theory in a case where bulk terms are in-
cluded by a scalar �eld. In our case we search a vector valued displace-
ment �eld which solves an elliptic system with in general discontinuous
coe�cients. Therefore, it is not clear how to conclude that minimisers
of the sharp interface functional are almost area minimising.
So far we discussed open problems with respect to the mathematical
analysis of phase separating systems. But also many other important
questions with fundamental importance for materials science and tech-
nological applications are still unanswered. Some of them are: In which
way do elastic interactions alter the coarsening process? In which cases
does coarsening slow down or even stop? What are circumstances in
which larger particles can split into several smaller ones? How do outer
forces in uence the size, shape and evolution of particles? For more
details on these issues and for an overview on modelling of phase sepa-
ration with elastic interactions we refer to Fratzl, Penrose and Lebowitz
[40].
80
8. Appendix
In this section we collect some known results used in the text.
Theorem 8.1. (Sobolev{Poincar�e inequality) There exists a con-
stant C(n; p) such that
�
�
Z
D
ju��
Z
D
uj
p
�
�
1
p
�
� C(n; p)(diamD)
�
�
Z
D
jruj
p
�
1
p
for all rectangles D � IR
n
and all u 2 W
1;p
(D). Here, p 2 (1;1),
p
�
=
np
n�p
and diamD is the diameter of D.
Proof. Without loss of generality we assume that
D = D
f
:=
n
n
X
i=1
x
i
e
i
j0 � x
i
� f
i
o
with f = (f
1
; :::; f
n
) and 0 < f
1
� ::: � f
n
: All other situations can
be reduced to one of these cases by a translation and a orthogonal
transformation.
The Sobolev{Poincar�e inequality
�
Z
D
e
jv ��
Z
D
e
vj
p
�
�
1
p
�
� C(n; p)
�
Z
D
e
jrvj
p
�
1
p
(108)
holds for all v 2 W
1;p
(D
e
) with a �xed constant C(n; p) where D
e
is the
unit cube, i.e. e = (1; :::; 1) 2 IR
n
. Now let D = D
f
and u 2 W
1;p
(D).
Then we de�ne
v(y) = u(f
1
y
1
; :::; f
n
y
n
) for all y 2 D
e
and obtain
rv(y) = (f
i
@
i
u(f
1
y
1
; :::; f
n
y
n
))
i=1;:::;n
: (109)
Hence
jrv(y)j
p
� f
p
n
jru(f
1
y
1
; :::; f
n
y
n
)j
p
:
Changing variables in (108) and using �
R
D
e
v = �
R
D
f
u we obtain
�
Z
D
ju��
Z
D
uj
p
�
(f
1
� ::: � f
n
)
�1
�
1
p
�
� C(n; p)f
n
�
Z
D
jruj
p
(f
1
� ::: � f
n
)
�1
�
1
p
:
Since L
n
(D) = f
1
� ::: � f
n
and since f
n
� diamD the theorem follows.
ut
81
Theorem 8.2. (Generalised Lebesgue convergence theorem) As-
sume E � IR
n
is measurable, g
n
�! g in L
q
(E) with 1 � q < 1 and
f
n
; f : E �! IR
n
are measurable functions such that
f
n
�! f a.e. in E ;
jf
n
j
p
� jg
n
j
q
a.e. in E
with 1 � p <1. Then f
n
�! f in L
p
(E) .
For a proof see [2].
Theorem 8.3. (Korn's inequality) Let be a bounded domain with
Lipschitz boundary.
i) There exists a constant c > 0 such that
Z
E(u) : E(u) � ckuk
2
H
1
for all u 2 X
2
:= fu 2 H
1
(; IR
n
) j (u;v)
H
1= 0 for all v 2 X
ird
g =
X
?
ird
where X
ird
:= fu 2 H
1
(; IR
n
) j there exist b 2 IR
n
and a skew
symmetric A 2 IR
n�n
such that u(x) = b+Ax g.
ii) Let � be an open subset of the boundary @ and let X
�
:= fu 2
H
1
(; IR
n
) ju
j�
= 0g.
Then there exists a constant c > 0 such that
Z
E(u) : E(u) � ckuk
2
H
1
for all u 2 X
�
:
A proof can be found for example in Zeidler [102].
Theorem 8.4. (Arzel�a{Ascoli theorem) Let B be a Banach space.
A subset F of C([0; T ];B) is relatively compact if and only if:
(i) ff(t) j f 2 Fg is relatively compact in B for all t 2 (0; T ),
(ii) F is uniformly equicontinuous, i.e. for all " > 0 there exists a
� > 0 such that
kf(t
2
)� f(t
1
)k
B
� "
for all f 2 F , and t
1
; t
2
2 [0; T ] such that jt
2
� t
1
j � �.
For a proof we refer to Simon [88].
Proposition 8.1. Let Q � IR
n
be a cube, g 2 L
q
loc
(Q) for a q > 1
and g � 0. Suppose that there exist a constant b > 0 and a function
f 2 L
r
loc
(Q) with r > q and f � 0 such that
�
Z
Q
R
(x
0
)
g
q
dx � b
�
�
Z
Q
2R
(x
0
)
g dx
�
q
+�
Z
Q
2R
(x
0
)
f
q
dx
for each x
0
2 Q and all R > 0 with 2R < dist(x
0
; @Q).
82
Then g 2 L
s
loc
(Q) for s 2 [q; q+ ") for some " > 0 and
�
�
Z
Q
R
(x
0
)
g
s
dx
�
1
s
� c
(
�
�
Z
Q
2R
(x
0
)
g
q
dx
�
1
q
+
�
�
Z
Q
2R
(x
0
)
f
s
dx
�
1
s
)
for all x
0
2 Q and R > 0 such that Q
2R
(x
0
) � Q. The positive
constants c and " depend on b, q, n and r.
For a proof of this proposition we refer to the book of Giaquinta [45]
or the paper of Giaquinta and Modica [46].
In the following we collect some facts on functions with bounded
variation and sets with bounded perimeter. For proofs and more details
we refer to the books of Evans and Gariepy [35] and Giusti [47].
De�nition 8.1. (Functions of bounded variation) Let � IR
n
be
open and u 2 L
1
(). Then we de�ne
Z
jruj := sup
�
Z
ur � g jg 2 C
1
0
(; IR
n
); jgj � 1
�
:
The set of functions of bounded variation is de�ned by
BV () :=
�
u 2 L
1
() j
Z
jruj <1
�
:
The set BV () is a Banach space with the norm
kuk
BV ()
:= kuk
L
1
()
+
Z
jruj:
Theorem 8.5. (Structure theorem for BV {functions) Let �
IR
n
and u 2 BV (). Then there exists a Radon measure � on and
a �{measurable function � : ! IR
n
such that
i) j�(x)j = 1 �{a.e.,
ii) �
R
ur � gdx =
R
g � � d�
for all g 2 C
1
0
(; IR
n
).
Remark 8.1. i) The above theorem shows that for functions u 2 BV ()
the gradient ru when understood in the sense of distributions can be
interpreted as a vector valued Radon measure. Precisely we get by de�n-
ing the Radon measures �
i
via
�
i
(D) =
Z
D
�
i
d� for Borel sets D �
that for all � 2 C
1
0
()
�
Z
u@
i
� =
Z
�d�
i
:
83
The fact that u 2 BV () implies that the vector valued Radon measure
(�
i
)
i=1;:::;n
has �nite total variation. We write
jruj = � and ru = (�
i
)
i=1;:::;n
;
i.e. we interpret jruj and ru as measures. In particular, it holds
Z
jruj = jruj() and �
Z
ur � g =
Z
g � ru
for all g 2 C
1
0
(; IR
n
).
ii) Let � IR
n
be open. Then the embedding
BV ()! L
1
()
is compact.
iii) The expression
R
jruj is lower semi-continuous with respect to L
1
{
convergence. For completeness we state this property more precisely.
Let � IR
n
be open and bounded and let (u
m
)
m2IN
be a sequence in
L
1
() such that
u
m
! u in L
1
() as m!1:
Then
Z
jruj � lim inf
m!1
Z
jru
m
j:
De�nition 8.2. (Sets of bounded perimeter) Let E � IR
n
be a
Borel set, X
E
its characteristic function and � IR
n
. Then we say
that E has �nite perimeter in if
Z
jrX
E
j <1:
Remark 8.2. i) For sets E with a smooth boundary
R
jrX
E
j mea-
sures the area of the part of the boundary of E that lie in , i.e.
Z
jrX
E
j = H
n�1
(@E \ ):
ii) For sets of bounded perimeter E in an open set � IR
n
we de�ne
the generalised outer unit normal
�
E
:= �
rX
E
jrX
E
j
:= ��
where � is the jrX
E
j{measurable function in the structure theorem.
We omit the index E in �
E
when it is clear from the context which set
we mean.
84
Theorem 8.6. (Co-area formula) Let u 2 BV (). Then it holds
Z
jruj =
Z
1
�1
Z
jrX
fu<tg
jdt:
De�nition 8.3. (Reduced boundary) Let E � IR
n
be a set with
locally �nite perimeter. We say that a point x 2 IR
n
belongs to the
reduced boundary @
�
E, if
(i) jrX
E
j(B
r
(x)) > 0 for all r > 0,
(ii)
lim
r!0
R
B
r
(x)
rX
E
R
B
r
(x)
jrX
E
j
exists and has length one.
Remark 8.3. For sets E � with �nite perimeter in the open set
it can be shown that
jrX
E
j(D) = H
n�1
(D \ @
�
E)
for all Borel sets D � .
De�nition 8.4. (Measure theoretic supremum) Let �
1
; :::; �
M
be
Radon measures de�ned on . Then we de�ne the measure theoretic
supremum
W
M
k=1
�
k
on all open sets D � by
M
_
k=1
�
k
!
(D) := sup
n
N
X
k=1
�
k
(B
k
) j B
k
� D; open, pairwise disjoint
o
:
Remark 8.4. i) The measure theoretic supremum
W
M
k=1
�
k
is the small-
est measure that dominates each of the measures �
k
on all Borel sets.
ii) Assume the M sequences of measures f�
l
k
g
l2IN
, k = 1; :::;M , ful�l
�
k
(D) � lim inf
l!1
�
l
k
(D) for all open D � . Then it holds
M
_
k=1
�
k
!
(D) � lim inf
l!1
M
_
k=1
�
l
k
!
(D)
for all open D � .
Theorem 8.7. Let X
k
2 BV (), k = 1; :::;M , de�ne a partition of the
open and bounded set , i.e. we require X
k
2 f0; 1g and
P
M
k=1
X
k
= 1
a.e.
Then it holds for all open sets
0
�
jrX
k
j(
0
) =
M
X
m=1;m6=k
H
n�1
(@
�
k
\ @
�
m
\
0
) ;
85
jr(X
k
+ X
l
)j(
0
)
=
N
X
m=1;m6=k;l
�
H
n�1
(@
�
k
\ @
�
m
\
0
) +H
n�1
(@
�
l
\ @
�
m
\
0
)
�
;
and for �
kl
:=
1
2
(jrX
k
j+ jrX
l
j � jr(X
k
+ X
l
)j) it holds
�
kl
(
0
) = H
n�1
(@
�
k
\ @
�
l
\
0
) :
In addition, it holds
rX
k
jrX
k
j
+
rX
l
jrX
l
j
= 0
and
r(X
k
+ X
l
)
jr(X
k
+ X
l
)j
= 0
�
kl
{almost everywhere.
For a proof see Vol'pert [91].
9. Notation
n space dimension,
C denotes any positive constant
depending on known quantities.
The value of C may change from
line to line in a given computation,
� IR
n
open, bounded domain with Lipschitz boundary,
L
n
n{dimensional Lebesgue measure,
jDj := L
n
(D) volume of the set D,
T arbitrary but �xed time in the existence proofs,
T
:= � (0; T ),
@ boundary of ,
n outer unit normal to ,
0
��
0
is compactly contained in the open set , i.e.
0
� and
0
is compact,
i; j; i
0
; j
0
indices varying between 1 and n,
IR
N�n
space of linear mappings from IR
n
into IR
N
,
we usually identify the space of linear mappings with the
N � n{matrices,
x = (x
i
)
i=1;:::;n
2 IR
n
coordinate of a point in physical space,
u = (u
i
)
i=1;:::;n
displacement vector,
86
ru =
�
@u
i
@x
j
�
i=1;:::;n;j=1;:::;n
displacement gradient,
E
ij
1
2
�
@u
i
@x
j
+
@u
j
@x
i
�
components of the linearized strain tensor,
E = (E
ij
)
i=1;:::;n;j=1;:::;n
strain tensor,
S = (S
ij
)
i=1;:::;n;j=1;:::;n
stress tensor,
t 2 [0;1) time,
N number of components,
k; l indizes varying between 1 and N ,
a � b eucledian inner product between vectors a;b in IR
n
and IR
N
,
A : B := tr(A
t
B) =
P
A
ij
B
ij
eucledian inner product in IR
n�n
and IR
N�n
,
A
t
transpose of a linear mapping,
Let A 2 IR
N�n
and d 2 IR
N
then we de�ne
A � d := A
t
d,
jAj :=
p
A : A eucledian norm of a tensor,
e
i
:= (�
ij
)
j=1;:::;n
i = 1; :::; n are the standard coordinate vectors,
� := fc
0
= (c
0
k
)
k=1;:::N
2 IR
N
j
P
N
k=1
c
0
k
= 1 g,
T� := fd = (d
k
)
k=1;:::N
2 IR
N
j
P
N
k=1
d
k
= 0 g tangent space to �,
P : IR
N
! T� euclidian projection onto T�, i.e.
Pd =
�
d
k
�
1
N
P
N
l=1
d
l
�
k=1;:::;N
for all d 2 IR
N
,
it holds: c �Pd = Pc � d for all c;d 2 IR
N
,
c = (c
k
)
k=1;:::;N
2 � vector of concentrations,
rc =
�
@c
k
@x
j
�
k=1;:::;N ;j=1;:::;n
concentration gradient,
E(c;u) =
R
�
1
2
rc : �rc+(c) +W (c; E) +W
�
(E)
free energy of a non{uniform system with elastic energy,
bulk chemical energy density at zero stress,
W elastic energy density,
W
k
elastic energy density of phase k,
used in Sections 5 and 6,
� temperature,
�
c
critical temperature,
W
�
(E) potential energy caused by external mechanisms,
typically we have
W
�
(E) = �E : S
�
with
S
�
= (S
�
ij
)
i=1;:::;n;j=1;:::;n
externally applied stress tensor,
which is assumed to be symmetric and independent of position,
� gradient energy tensor, a constant linear mapping of
IR
N�n
into itself,
87
Assumptions: �A : B = A : �B
and A : �A �
0
jAj
2
with a
0
> 0,
trA trace of a linear mapping of a �nite dimensional
vector space into itself,
" scaling parameter in the free energy
used in Sections 5 and 6,
�
;z
partial (or sometimes total derivative) of a function �
with respect to a scalar, vector or tensor variable z,
in particular we have:
;c
= (
;c
k
)
k=1;:::;N
=
�
@
@c
k
�
k=1;:::;N
,
W
;c
= (W
;c
k
)
k=1;:::;N
=
�
@W
@c
k
�
k=1;:::;N
,
W
;E
=
�
W
;E
ij
�
i=1;:::;n;j=1;:::;n
=
�
@W
@E
ij
�
i=1;:::;n;j=1;:::;n
,
a particular form of W is
W (c
0
; E
0
) =
1
2
(E
0
� E
?
(c
0
)) : C(c
0
) (E
0
� E
?
(c
0
)),
E
?
(c
0
) stress free strain for concentrations c
0
2 �
(assumed to be a symmetric tensor),
C(c
0
) = (C
iji
0
j
0
(c
0
))
i;j;i
0
;j
0
=1;:::;n
elasticity tensor,
which maps symmetric tensors into symmetric tensors
by the de�nition (CE
0
)
ij
=
P
i
0
;j
0
C
iji
0
j
0
E
0
i
0
j
0
,
we require:
C
iji
0
j
0
= C
ijj
0
i
0
= C
jii
0
j
0
,
We assume:
C is symmetric, i.e. CE
0
: F
0
= E
0
: CF
0
This implies:
C
iji
0
j
0
= C
i
0
j
0
ij
C is positive de�nite uniformly in c
0
, .i.e.
there exists a positive constant c
2
> 0 sucht that
E
0
: CE
0
� c
2
jE
0
j
2
,
C is bounded uniformly in c
0
,
C
;c
is bounded uniformly in c
0
,
It holds:
W
;c
=
�
�E
?
;c
i
(c) : C(c) (E
0
� E
?
(c))
+
1
2
(E
0
� E
?
(c)) : C
;c
i
(c) (E
0
� E
?
(c)))
i=1;:::;N
For abbreviation we write:
W
;c
= �E
?
;c
(c) : C(c) (E � E
?
(c))
+
1
2
(E � E
?
(c)) : C
;c
(c) (E � E
?
(c))
r � f divergence operator of f : ! IR
n
, i.e. r � f = trrf
div
T
surface divergence on the surface T ,
r � S for S : ! IR
n
1
�n
is de�ned as
88
r � S := (r � S
i
)
i=1;:::;n
1
where S
i
= (S
ij
)
j=1;:::;n
and S = (S
ij
)
i=1;:::;n
1
;j=1;:::;n
with this notation it holds
R
r � S =
R
@
SndH
n�1
,
D
0
() distributions on ,
L
p
(; IR
N
) For spaces of functions we use the standard de�nitions
W
m;p
(; IR
N
) and notations. For details we refer to the books of
W
m;p
0
(; IR
N
) Alt [2], Evans [34], and Evans and Gariepy [35],
H
m
(; IR
N
) = W
m;2
,
L
p
(0; T ;X) X Banach space,
W
m;p
(0; T ;X)
H
m
(0; T ;X)
L
p
() := L
p
(; IR),.....
when it is clear from the context we also sometimes write L
p
()
instead of L
p
(; IR
N
),
Z
?
the subspace orthogonal to Z,
X
�
dual space of a Banach space X,
X
ird
:= fu 2 H
1
(; IR
n
) j there exist b 2 IR
n
and
a skew symmetric A 2 IR
n�n
such that u(x) = b+Ax g,
which is the space of in�nitesimal rigid motions,
i.e. translations and in�nitesimal rotations,
X
1
:= fc 2 H
1
(; IR
N
) j c 2 � almost everywhere g,
X
2
:= fu 2 H
1
(; IR
n
) j (u;v)
H
1= 0 for all v 2 X
ird
g = X
?
ird
,
X
m
1
:= fc 2 X
1
j �
R
c
k
= m
k
g
where m = (m
k
)
k=1;:::;N
is a constant vector of mean values,
Y := fz 2 H
1
(; IR
N
) j
R
z = 0;
P
N
k=1
z
k
= 0 g,
D := ff 2
�
H
1
(; IR
N
)
�
�
j hd; fi = 0
for all d = d(x)(1; :::; 1)
t
with a scalar valued function
d 2 H
1
() and all d � e
k
; k = 1; :::; N g,
which is the space of all functionals
being zero on the L
2
{complement of Y ,
(:; :) inner product in a Hilbert space,
hu; fi duality pairing of an element u of a Banach space
and an element f lying in the dual of the Banach space,
i.e. hu; fi = f(u),
J
k
= (J
ki
)
i=1;:::;n
ux of component k 2 f1; :::; Ng,
J = (J
ki
)
k=1;:::;N ;i=1;:::;n
,
L = (L
kl
)
k=1;:::;N ;l=1;:::;N
mobility matrix,
@
t
partial derivative with respect to time,
�
k
chemical potential of component k,
� = (�
k
)
k=1;:::;N
vector of chemical potentials,
89
w = (w
k
)
k=1;:::;N
vector of generalised chemical potential di�erences,
� = (�
k
)
k=1;:::;N
vector of the mean values of the chemical potentials,
m = (m
i
)
i=1;:::;N
vector of mean values of the concentrations,
0 N{vector where each component is zero,
M number of time intervals in the implicit time discretisation,
�t = T=M time step,
m index for the time discrete solutions,
E
m;�t
time discrete energy with time step �t at time m�t,
(c
m
;w
m
;u
m
)time discrete solution at time m�t,
� test function in the weak formulation of the concentrations,
� test function in the weak formulation of the chemical potentials,
� test function in the weak formulation of the displacement,
c
0
2 X
1
initial data for the concentrations,
D
c
E Gateaux derivative of E with respect to the variable c,
D
u
E Gateaux derivative of E with respect to the variable u,
L the mapping which maps w to L�w as a mapping from Y to D,
G inverse of the mapping L (generalised Green operator),
N subset of IN, used for labelling subsequences,
we will use the same notation for di�erent subsequences,
k
B
Boltzmann constant,
A (A
ij
)
i=1;:::;N ;j=1;:::;N
interaction parameters,
� regularising parameter in Section 4,
G := fd 2 IR
N
j
P
N
i=1
d
k
= 1 and d
k
� 0 for k = 1; :::; Ng
the Gibbs simplex,
B
R
(x) ball of radius R with centre x,
Q
R
(x
0
) := fx 2 IR
n
j jx
i
� x
0i
j < Rg
with x
0
2 IR
n
, R > 0,
�
kl
:= @
�
k
\ @
�
l
\
interface between the phases k and l,
� (n� 2){dimensional manifold describing multiple junctions,
�
kl
surface tension of the interface between the phases k and l,
R
jruj total variation of the distributional derivative of u,
jruj(E) :=
R
E
jruj total variation measure,
X
E
characteristic function of a set E,
�
E
:= �
rX
E
jrX
E
j
,
generalised outer unit normal,
� variation of the independent variable (in Section 5),
� deformations of the set ,
References
[1] F. Almgren, Existence and Regularity Almost Everywhere of Elliptic Varia-
tional Problems with Constraints, Memoirs A.M.S. 165 (1976).
90
[2] H. W. Alt, Lineare Funktionalanalysis, Springer Verlag, Berlin (1985).
[3] H. W. Alt and S. Luckhaus, Quasilinear elliptic parabolic di�erential equa-
tions, Math. Z., 183 (1983), pp. 311{338.
[4] H. W. Alt and I. Pawlow, A mathematical model of dynamics of non{
isothermal phase separation, Physica D, 59 (1992), pp. 389{416.
[5] H. W. Alt and I. Pawlow, Existence of solutions for non{isothermal phase
separation, Adv. Math. Sci. Appl., 1 (1992), pp. 319{409.
[6] H. W. Alt and I. Pawlow, Models of non{isothermal phase transitions in
multicomponent systems. Part I: Theory, SFB256 University Bonn, preprint
222 (1992).
[7] L. Ambrosio,Metric space valued functions of bounded variation, Ann. Scuola
Normale Superiore Pisa, (4) 17 (1990), pp. 439{478.
[8] S. Baldo, Minimal interface criterion for phase transitions in mixtures of
Cahn{Hilliard uids, Ann. I. H. P. analyse nonlin�eaire, 7 (1990), pp. 37{65.
[9] S. Baldo and G. Belletini, �{convergence and numerical analysis: an
application to the minimal partition problem, Ric. di Matematica, Vol. XL
(1991), pp. 33{64.
[10] J.W. Barrett and J.F. Blowey, An error bound for the �nite element
approximation of the Cahn{Hilliard equation with logarithmic free anergy, Nu-
mer. Math., 72 (1995), pp. 257{287.
[11] J.W. Barrett and J.F. Blowey, An error bound for the �nite element
approximation of a model for phase separation of a multi{component alloy,
IMA J. Numer. Anal., 16 (1996), pp. 257{287.
[12] J.W. Barrett, J.F. Blowey, and H. Garcke, Finite element approxima-
tion of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Num.
Anal., Vol 37, No.1, pp. 286{318, (1999).
[13] J.W. Barrett, J.F. Blowey, and H. Garcke, On fully practical �nite ele-
ment approximations of degenerate Cahn{Hilliard systems, SFB256 University
Bonn, preprint.
[14] J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for
phase separation with non-smooth free energy Part I: Mathematical analysis
European J. Appl. Math., 2 (1991), pp. 147{179.
[15] J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for
phase separation with non-smooth free energy Part II: Numerical analysis. Eu-
ropean J. Appl. Math., 3 (1992), pp. 147{179.
[16] L. Bronsard, H. Garcke, and B. Stoth, A multi{phase Mullins{Sekerka
system: matched asymptotic expansions and an implicit time discretisation for
the geometric evolution problem, Proc. Roy. Soc. Edinburgh, 128 A (1998), pp.
481{506.
[17] J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), pp. 795{801.
[18] J. W. Cahn, On spinodal decomposition in cubic crystals, Acta Metall., 10
(1962), pp. 179{183.
[19] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I.
Interfacial free energy, Journal of Chemical Physics, Vol. 28 (1958), pp. 258{
267.
[20] J. W. Cahn and F. C. Larch
�
e, Surface stress and the chemical equilibrium
of single crystals II: solid particles imbedded in a solid matrix, Acta Metall.,
30 (1982), pp. 51{56.
91
[21] J. W. Cahn and F. C. Larch
�
e, A linear theory of thermomechanical equi-
librium of solids under stress, Acta Metall., 21 (1973), pp. 1051{1063.
[22] P. G. Ciarlet, Mathematical Elasticity, North Holland, Amsterdam, New
York, Oxford (1988).
[23] M.I.M. Copetti and C.M. Elliott, Numerical analysis of the Cahn-
Hilliard equation with logarithmic free energy, Numer. Math. 63 (1992), pp.
39{65.
[24] A. Debussche and L. Dettori, On the Cahn{Hilliard equation with a log-
arithmic free energy, Nonlinear Analysis, Theory, Methods and Appl., 24 No.
10 (1995), pp. 1491{1514.
[25] D. De Fontaine, An analysis of clustering and ordering in multicomponent
solid solutions{I. stability criteria, Journal Phys. Chem. Solids, 33 (1972), pp.
287{310.
[26] D. De Fontaine, An analysis of clustering and ordering in multicomponent
solid solutions{II. uctuations and kinetics, Journal Phys. Chem. Solids 34
(1973), pp. 1285{1304.
[27] W. Dreyer and W. H. M
�
uller, A study of the coarsening in tin/lead
solders, International Journal of Solids and Structures, in print.
[28] C.M. Elliott, The Cahn-Hilliard model for the kinetics of phase transitions,
in Mathematical Models for Phase Change Problems, J.F. Rodrigues (ed.),
International Series of Numerical Mathematics, Vol. 88, Birkh�auser-Verlag,
Basel, pp. 35{73.
[29] C.M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degen-
erate mobility, SIAM J. Math. Anal., 27 (1996), pp. 404{423.
[30] C.M. Elliott and H. Garcke, Di�usional phase transitions in multicom-
ponent systems with a concentration dependent mobility matrix, Physica D,
109 (1997), pp. 242 - 256.
[31] C. M. Elliott and S. Luckhaus A generalised di�usion equation for phase
separation of a multi{component mixture with interfacial free energy, SFB256
University Bonn, Preprint 195 (1991).
[32] C. M. Elliott and Zheng Songmu, On the Cahn{Hilliard equation, Arch.
Rat. Mech. Anal., 96 (1986), pp. 339{357.
[33] J. D. Eshelby, Elastic inclusions and inhomogeneities, Prog. Solid Mech., 2
(1961), pp. 89{140.
[34] L. C. Evans, Partial Di�erential Equations, AMS,Graduate Studies in Math-
ematics Vol. 19, Providence (1997).
[35] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of
Functions, CRC Press, Boca Raton, Ann Arbor, London (1992).
[36] D. J. Eyre, Systems of Cahn{Hilliard equations, SIAM J. Appl. Math., 53
(1993), 1686{1712.
[37] H. Federer, Geometric Measure Theory, Springer, Berlin, Heidelberg, New{
York (1969).
[38] I. Fonseca and L. Tartar, The gradient theory of phase transitions for
systems with two potential wells, Proc. Roy. Soc. Edinburgh, 111 A (1989), pp.
89{102.
[39] P. Fratzl, J. L. Lebowitz, O. Penrose, and J. Amar, Scaling functions,
self similarity, and the morphology of phase separating systems, Phys. Rev. B,
44 (1991), pp. 4794{4811.
92
[40] P. Fratzl, O. Penrose, and J. L. Lebowitz,Modelling of phase separation
in alloys with coherent elastic mis�t, J. Stat. Physics, 95 5/6 (1999), pp. 1429{
1503.
[41] H. Garcke and A. Novick{Cohen, A singular limit for a system of degen-
erate Cahn{Hilliard equations, Adv. in Di�. Equations (to appear).
[42] H. Garcke, M. Rumpf, and U. Weikard, The Cahn-Hilliard equation with
elasticity: Finite element approximation and qualitative studies, submitted.
[43] H. Garcke and T. Sturzenhecker, The degenerate multiphase Stefan
problem with Gibbs-Thomson law, Adv. Math. Sci. Appl., 8 (1998), pp. 929{
941.
[44] F. W. Gehring, The L
p
{integrability of the partial derivatives of a quasi
conformal mapping, Acta Math., 130 (1973), pp. 265{277.
[45] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear
elliptic systems, Annals of Mathematics Studies, Princeton University Press,
(1983).
[46] M. Giaquinta and G. Modica, Regularity results for some classes of higher
order nonlinear elliptic systems, J. f�ur reine u. angew. Math., 311/312 (1979),
145{169.
[47] E. Giusti,Minimal Surfaces and Functions of Bounded Variation, Birkh�auser
Verlag, Basel, Boston, Stuttgart, (1984).
[48] M. E. Gurtin, An Introduction to Continuum Mechanics, Academic Press,
Boston, New York, London (1981).
[49] M. E. Gurtin, On a nonequilibrium thermodynamics of capillarity and phase,
Quart. Appl. Math., 47 (1989), pp. 129{145.
[50] M. E. Gurtin, Generalised Ginzburg{Landau and Cahn{Hilliard equations
based on a microforce balance, Physica D, 92 (1996), pp. 178{192.
[51] M. E. Gurtin and P. W. Voorhees, The continuum mechanics of coherent
two-phase elastic solids with mass transport, Proc. Royal Soc. London A, 440
(1993), pp. 323{343.
[52] J. J. Hoyt, Spinodal decomposition in ternary alloys, Acta Metallurgica 37
(1989), pp. 2489{2497.
[53] J. J. Hoyt, Linear spinodal decomposition in a regular ternary alloy, Acta
Metallurgica 38 (1990), pp. 227{231.
[54] J. J. Hoyt, The continuum theory of nucleation in multicomponent systems,
Acta Metallurgica 38 (1990), pp. 1405{1412.
[55] W. C. Johnson and J. J. D. Alexander, Interfacial conditions for ther-
momechanical equilibrium in two{phase crystals, J. Appl. Phys. 59 (1986), pp.
2735{2746.
[56] W. C. Johnson and J. W. Cahn, Elastically induced shape bifurcations of
inclusions, Acta Metall., 32 (1985), pp. 1935{1953.
[57] M. J. Kaufman, P. W. Voorhees, W. C. Johnson and F. S. Bian-
caniello, An elastically induced morphological instability of a mis�tting pre-
cipitate, Metall. Trans., 20A (1989), 2171{2175.
[58] A. G. Khachaturyan, Some questions concerning the theory of phase transi-
tions in solids, Fiz. Tverd. Tela., 8 (1966), pp. 2709{2717. English translation
in Sov. Phys. Solid. state 8 (1966), pp. 2163.
[59] A. G. Khachaturyan, Theory of Structural Transformations in Solids, Wi-
ley, New York (1983)
93
[60] J. S. Kirkaldy and D. J. Young, Di�usion in the Condensed State, The
Institute of Metals, London (1987).
[61] F. Langmayr, P. Fratzl, and G. Vogl, Crossover from !{phase to �{
phase precipitation in bcc Ti{Mo, Phys. Rev. B, 49 (1994), pp. 11759{11766.
[62] F. C. Larch
�
e and J. W. Cahn, Thermochemical equilibrium of multiphase
solids under stress, Acta Metall., 26 (1978), pp. 1579{1589.
[63] F. C. Larch
�
e and J. W. Cahn, The e�ect of self{stress on di�usion in
solids, Acta Metall., 30 (1982), pp. 1835{1845.
[64] F. C. Larch
�
e and J. W. Cahn, Phase changes in a thin plate with non{local
self{stress e�ects, Acta Metall., 40 (1992), pp. 947{955
[65] P. H. Leo, J. S. Lowengrub, and H. J. Jou, A di�use interface model for
microstructural evolution in elastically stressed solids, Acta Mater., 46 (1998),
pp. 2113{2130.
[66] P. H. Leo and R. F. Sekerka, The e�ect of surface stress on crystal{melt
and crystal{crystal equilibrium, Acta Metall., 37 (1989), pp. 3119{3138.
[67] P. H. Leo, W. W. Mullins, R. F. Sekerka, and J. Vinals, E�ects of
elasticity on late stage coarsening, Acta Metall., 38 (1990), pp. 1573{1580.
[68] I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from su-
persaturated solid solutions, J. Phys. Chem. Solids, 19 (1961), pp. 35{50.
[69] F.{H. Lin, Variational problems with free interfaces, Calc. Var., 1 (1993), pp.
149{168.
[70] L. L
�
ochte, Real{Zeit skalierte Modellierung der Bildung und des Wachs-
tums von Guinier{Preston{Zonen in AlCu{Legierungen, PhD thesis, RWTH
Aachen.
[71] S. Luckhaus, The Stefan problem with the Gibbs-Thomson relation for the
melting temperature, Europ. J. Appl. Math., 1 (1991), pp. 101{111.
[72] S. Luckhaus, Solidi�cation of alloys and the Gibbs-Thomson law, Preprint
No. 335, SFB 256 Universit�at Bonn (1994).
[73] S. Luckhaus and L. Modica, The Gibbs{Thomson relation within the gra-
dient theory of phase transitions, Arch. Rat. Mech. Anal., 107 (1989), pp.
71{83.
[74] S. Luckhaus and T. Sturzenhecker, Implicit time discretisation for the
mean curvature ow equation, Calc. Var., 3 (1995), pp. 253{271.
[75] T. Miyazaki and M. Doi, Shape bifurcations in the coarsening of precipitates
in elastically constrained systems, Mater. Sci. Engng A, 110 (1989), pp. 175{
185.
[76] T. Miyazaki, M. Doi and T. Kozakai, Shape bifurcations in the coarsening
of precipitates in elastically constrained systems, Solid State Phenomena, 3
(1988), pp. 227{237.
[77] L. Modica, The gradient theory of phase transition and the minimal interface
criterion, Arch. Rat. Mech. Anal., 98 (1987), pp. 123{142.
[78] L. Modica, The gradient theory of phase transition with boundary contact
energy, Ann. Inst. H. Poincar�e, Anal. nonlin., 4 (1987), pp. 487{512.
[79] L. Modica and S. Mortola, Un esempio di �{convergenza, Boll. Un. Mat.
Ital., 5 14{B (1977), pp. 285{299.
[80] F. Morgan, Geometric Measure Theory: A beginners guide, Academic Press,
New York, London, Tokyo (1995).
[81] H. Nishimori and A. Onuki, Pattern formation in phase{separating alloys
with cubic symmetry, Phys. Rev. B, 42 (1990), pp. 980{983.
94
[82] H. Nishimori and A. Onuki, Freezing of domain growth in cubic solids with
elastic mis�t, J. Phys. Soc. Jpn., 60 (1991), pp. 1208{1211.
[83] Nishimori and A. Onuki, Anomalously slow domain growth due to a modulus
inhomogeneity in phase{separating alloys, Phys. Rev. B, 43 (1991), pp. 13649{
13652.
[84] A. Novick{Cohen, The Cahn-Hilliard equation: mathematical and modelling
perspectives. Adv. Math. Sci. Appl., 8 (1998), pp. 965{985.
[85] L. Onsager, Reciprocal relations in irreversible processes I, Phys. Rev. 37
(1931), pp. 405{426.
[86] L. Onsager, Reciprocal relations in irreversible processes II, Phys. Rev. 38
(1931), pp. 2265{2279.
[87] A. Onuki, Ginzburg{Landau approach to elastic e�ects in the phase separation
of solids, J. Phys. Soc. Jpn., 58 (1989), pp. 3065{3068.
[88] J. Simon, Compact sets in the space L
p
(0; T ;B), Annali di Mathematica Pura
ed Applicata 146 (1987), pp. 65{96.
[89] P. Sternberg, Vector{valued local minimisers of nonconvex variational prob-
lems, Rocky Mountain J. Math. 21 (1991), pp. 799{807.
[90] R. Temam, Navier{Stokes equations, North Holland, Amsterdam, New York,
Oxford (1977).
[91] A. I. Vol'pert, The spaces BV and quasilinear equations, Math. USSR{
Sbornik, 2 No. 2 (1969), pp. 225{267.
[92] A. I. Vol'pert and S.I. Huhjaev, Analysis in classes of discontinuous func-
tions and equations of mathematical physics, Martinus Nijho� Publisher, Dor-
drecht, (1985).
[93] P. W. Voorhees, Ostwald ripening of two phase solids, A. Rev. Mater. Sci.,
22 (1992), pp. 197{215.
[94] P. W. Voorhees, G. B. McFadden, and W. C. Johnson, On the mor-
phological development of second phase particles in elastically stressed solids,
Acta Metall., 40 (1992), pp. 2979{2992.
[95] J. D. van der Waals, The thermodynamic theory of capillarity under the
hypothesis of a continuous variation of density (in Dutch), Verhaendel. Kronik.
Akad. Weten. Amsterdam, Vol. 1, (1893), Engl. translation by J.S. Rowlinson,
J. Stat. Phys., 20 (1979), pp. 197{244.
[96] C. Wagner, Theorie der Alterung von Niederschl�agen durch Uml�osen, Z.
Elektrochem., 65 (1961), pp. 581{594.
[97] W. v. Wahl, On the Cahn{Hilliard equation u
0
+ �
2
u � �f(u) = 0, Delft
Progress Report, 10 (1985), pp. 291{310.
[98] Y. Wang, L. Q. Chen, and A. G. Khachaturyan, Modelling o dynamical
evolution of micro/mesoscopic morphological patterns in coherent phase trans-
formations, H.O. Kirchner et al. (eds.), Computer Simulations in Materials
Science, Kluwer Academic Publishers, Netherlands (1996), pp. 325{371.
[99] U. Weikard, Numerische L�osungen der Cahn{Hilliard{Gleichung unter Ein-
beziehung elastischer Materialeigenschaften, diploma thesis, University Bonn,
(1998).
[100] T. Young, An essay on the cohesion of uids, Philos. Trans. Roy. Soc.
London, 95 (1805), 65{87.
[101] E. Zeidler, Nonlinear Functional Analysis and its Applications II b),
Springer, New{York, Berlin, Heidelberg (1990).
95
[102] E. Zeidler, Nonlinear Functional Analysis and its Applications IV, Springer,
New{York, Berlin, Heidelberg (1988).
96
