Journal of Convex Analysis
Volume 7 (2000), No. 2, 209–241
A Twodimensional Variational Model for the
Equilibrium Configuration of an Incompressible,
Elastic Body with a Three-Well Elastic Potential
Martin Fuchs
Universität des Saarlandes, Fachbereich 9 Mathematik,
Postfach 15 11 50, 66041 Saarbrücken, Germany.
e-mail: fuchs@math.uni-sb.de
Gregory Seregin
V. A. Steklov Mathematical Institute, St. Petersburg Branch
191011 St. Petersburg, Russia.
e-mail: seregin@pdmi.ras.ru
Received June 15, 1999
We consider a geometrically linear variational model in two space dimensions for an incompressible,
elastic body whose elastic potential has exactly three wells corresponding to one austenitic and to two
martensitic phases. Passing to the dual problem we show that the stress tensor is weakly differentiable
on the interior of the domain and in addition Hölder continuous on any subset of the union of the pure
phases.
1991 Mathematics Subject Classification: 73C05, 73V25, 73G05
1.
Introduction
In this paper we are concerned with the mathematical analysis of the variational problem
which corresponds to the physical situation that a multiphase elastic body is in equilibrium
state under the action of a given system of forces. In order to obtain such a variational
formulation, we have to assume that the temperature as well as the loads are fixed. We
are also not going to consider the most general situation which means that we restrict
ourselves to the geometrically linear case, for a description of the nonlinear setting and
its comparison with the linear one we refer the reader to [4, 5], [7] and [12, 13]. Another
restriction is that we consider an elastic potential with three wells corresponding to one
austenitic and to two martensitic phases. Even under these assumptions the analysis
of the problem turns out to be quite difficult for the following reason: by definition the
elastic energy is the pointwise infimum of the different phase energies and so in general not
quasiconvex. Hence one has to consider the quasiconvex envelope for which Dacorogna’s
formula (see [8]) is available. Unfortunately this representation is not very explicit and
so only very few examples for the computation of the envelope are known, in particular
this concerns the case of three wells. For two wells with the same elastic moduli explicit
formulas for the quasiconvex envelope can be found in the papers [13], [16] and [19],
the case of two isotropic wells with well-ordered elastic moduli is solved in principle in
[1, 2] which means that the problem is reduced to the minimization of suitable functions
depending only on a finite number of variables. We wish to mention that for incompressible
ISSN 0944-6532 / $ 2.50
c Heldermann Verlag
­
210
M. Fuchs, G. Seregin / A twodimensional variational model
bodies there is some analogue of Dacorogna’s formula obtained in [22] (see also [18] for
a more general setting), and in the case of two isotropic wells explicit representations of
the quasiconvex envelope are given in [14] and [22].
For completeness we would like to mention that in the paper [13, Section 8, p. 232] the
reader will find some comments concerning the computation of the envelope in the case
of N -wells but no explicit formula is given.
The importance of explicit formulas is evident: as a matter of fact one should expect
degeneracy of the relaxed functional, and since Dacorogna’s formula is not local, it is
quite difficult to decide where degeneracy occurs and what the precise behaviour of the
energy is.
However, there are several cases for which the quasiconvex envelope turns out to be a
convex or “almostÔ convex function, e.g. the case of two wells with the same elastic
moduli. Here the quasiconvex envelope is convex provided the stress-free strains are
compatible (see [13] for details), or it can be replaced by a convex integrand in the case
of incompatible stress-free strains (see [21]). It is then possible to apply the powerful
methods of duality theory (compare [20] and [21]) to this particular situation.
Another setting recently has been studied by the second author in [22]: this paper addresses the case of incompressible bodies in two spatial variables and it is shown that the
notions of convexity and quasiconvexity coincide. Motivated by this result we also impose
incompressibility as a further restriction, and we will limit ourselves to the twodimensional
case. For two space dimensions one may also use an alternative approach for the construction of a suitable relaxed variational problem. If, for example, Ω is simply connected,
then we may look for solutions of our original variational problem which are the “curlÔ
of some scalar function, thus we arrive at a scalar variational problem of higher order.
For the relaxation of first order scalar problems we refer to [9], the case of higher order
problems is treated in [18]. Besides the topological constraint it is also not immediate
how to incorporate the boundary conditions in this setting.
Our paper is organized as follows: In Section 2 we fix our notation and introduce the
basic variational problem together with its relaxation. We also formulate the natural
dual variational problem whose unique solution σ has the meaning of the stress tensor.
Lemma 2.1 contains the so-called effective stress-strain relation, in Theorem 2.2 we show
weak differentiability of σ, and our main result Theorem 2.4 states that σ is Hölder
continuous on any region where no microstructure occurs. In Section 3 Lemma 2.1 is
established, in Section 4 we prove Theorem 2.2. The proof of Theorem 2.4 presented in
Section 7 is based on some local estimates of Caccioppoli-type (see Section 5) and on a
decay lemma for the squared mean oscillation of σ (see Section 6) being valid at centers
x where the formation of microstructure is excluded.
Acknowledgements.
This work was done during the second author’s stay at the MaxPlanck Institut, Leipzig. He would like to thank S. Müller for many stimulating discussions.
The second author was partially supported by INTAS, grant No. 96-835, and by RFFI, grant No.
96-01-00824. The final version was completed during the first author’s visit at the University of
Helsinki. He acknowledges support by the Academy of Finland.
The authors would like to thank the referee for his valuable comments.
M. Fuchs, G. Seregin / A twodimensional variational model
2.
211
Formulation of the problem and statement of the main results
Let Md denote the space of all real d × d matrices. Sd is the subspace consisting of
symmetric matrices. We will use the following notation
√
u · v = ui vi , |u| = u · u,
u ⊗ v = (ui vj ) ∈ Md ,
1
u¬v =
(u ⊗ v + v ⊗ u) ∈ Sd for u = (ui ), v = (vi ) ∈ Rd ,
2
√
A : B = tr AT B = Aij Bij , AT = (Aji ) ∈ Md , |A| = A : A,
Aa = (Aij aj ) ∈ Rd for A = (Aij ), B = (Bij ) ∈ Md , a ∈ Rd ,
where the convention of summation over repeated Latin indices running from 1 to d is
adopted. Further we let
◦
◦
◦
¨
©
Md = A ∈ Md : tr A = 0 , Sd = Sd ∩ Md
◦
◦
◦
◦
M = M2 , M = M2 , S = S2 and S = S2 .
We consider an elastic body with three different phases, an austenitic one and two martensitic phases. Moreover, we restrict ourselves to the twodimensional case (d = 2) and assume in addition that the body is incompressible. The energy density (or elastic potential
or just energy) of the austenitic (the 1st) phase is given by
◦
g1 (ε) = µ|ε|2 + g0 , ε ∈ S,
where µ is a positive constant, and g0 denotes a constant depending on the temperature
T . For the martensitic phases the densities are of the form
◦
g2 (ε) = |ε − ε0 |2 , g3 (ε) = |ε + ε0 |2 , ε ∈ S;
here ε0 and −ε0 are the stress-free strains of the martensitic phases. In this setting it is
assumed that the energies of the martensitic phases have the same minima and that the
common value is equal to zero. Of course this is no further restriction since we may add
any constant to the energies. On the other hand it follows from our notation that the
elastic moduli of the martensitic phases are the same, and just for simplicity we put them
equal to 1. As mentioned above the constant g0 depends on the temperature, precisely
we have


g0 > 0 if T < T0 ,
g0 = 0 if T = T0 ,


g0 < 0 if T > T0
which means that for T > T0 the stress-free state of the austenitic phase is preferred
whereas for T < T0 the stress-free states of the martensitic phases are favoured. Here T0
denotes the transition temperature.
Now the energy density of this three-phase body is given by
212
M. Fuchs, G. Seregin / A twodimensional variational model
◦
g(ε) = min{g1 (ε), g2 (ε), g3 (ε)}, ε ∈ S,
and according to the general theory the state of phase equilibrium is described in terms
of the following variational problem:
Problem P:
◦
Find a vector-valued function u ∈ J 12 (Ω) + u0 such that
◦
©
¨
I(u) = inf I(v) : v ∈ J 12 (Ω) + u0 .
Here Ω is a bounded Lipschitz domain in R2 (the undeformed state of the body is represented by the set Ω); we further let (ε(v) denoting the symmetric derivative)
I(v) =
Z
(g(ε(v)) − f · v) dx
Ω
and assume
f ∈ L2 (Ω; R2 ) ,
u0 ∈ J21 (Ω).
(2.1)
As usual Lp (Ω; R2 ) denotes the Lebesgue space of all vectorfields from Ω into R2 being pintegrable, the Sobolev space Wp1 (Ω; R2 ) is defined as the subspace of Lp (Ω; R2 ) consisting
of those fields whose first weak derivatives are generated by Lp -functions. Finally, we let
Jp1 (Ω) = {v ∈ Wp1 (Ω; R2 ) : div v = 0 on Ω},
◦
•
J 1p (Ω) = closure of C ∞ (Ω) in Wp1 (Ω; R2 ),
•
C ∞ (Ω) = {v ∈ C0∞ (Ω; R2 ) : div v = 0 on Ω}.
The energy density g is continuous and bounded from below. Moreover, there exist
constants ν > 0 , c1 , c2 ≥ 0 such that
ν|ε|2 − c1 ≤ g(ε) ≤
◦
1 2
|ε| + c2
ν
(2.2)
is true for any ε ∈ S. Clearly estimate (2.2) combined with Korn’s inequality implies
boundedness of any minimizing sequence in the space J21 (Ω), and one may pass to weakly
convergent subsequences. But unfortunately our functional I is not sequentially weakly
lower semicontinuous on the space J21 (Ω) and examples show that in fact problem P may
fail to have solutions. The question of weak lower semicontinuity for functionals like our
energy I defined on the space J21 (Ω) has been investigated in the paper [22] with the result
that this property of the energy is (under some additional assumptions on the density)
equivalent to J21 - quasiconvexity of the integrand. This notion is a natural analogue
of the definition of quasiconvexity introduced by Morrey [17] (or the concept of Wp1 quasiconvexity due to Ball and◦ Murat [6]) for the case of solenoidal vectorfields. We say
that a continuous function h : S → R is Jp1 - quasiconvex iff
M. Fuchs, G. Seregin / A twodimensional variational model
Z
h(A) dx ≤
ω
Z
213
h (A + ε(u)) dx
ω
◦
◦
holds for any A ∈ S, for any bounded open set ω ⊂ R2 and for all functions u ∈ J 1p (ω).
Clearly we have the same definition for any dimension d ≥ 2.
Since problem P has in general no solution, we now pass to a suitable relaxed variational
problem, i.e. we look at
Problem QP:
◦
Find a function u ∈ u0 + J 12 (Ω) such that
◦
QI(u) = inf{QI(v) : v ∈ u0 + J 12 (Ω)}.
Here the relaxed energy is given by the formula
Z
QI(v) = (Qg(ε(v)) − f · v) dx
Ω
and Qg denotes the J21 -quasiconvex envelope of g for which in [22] the following representation formula has been established (compare also [18] for a more general setting):



Z
◦
1
Qg(κ) = inf − g(κ + ε(v)) dx : v ∈ J 2 (B) ,


B
◦
κ ∈ S , B = {x ∈ R2 : |x| < 1}.
Moreover, the next statements are also due to [22]:
•
•
•
Problem QP has at least one solution.
Any weak limit of any subsequence of a minimizing sequence of problem P is termed
a general solution of problem P. These generalized solutions are exactly the solutions
of QP.
For our particular integrand we have Qg(κ) = g ∗∗ (κ), κ ∈ S, g ∗∗ denoting the second
Young transform, hence Qg is a convex function.
Let us recall the definitions
◦
◦
g ∗∗ (κ) = sup{κ : τ − g ∗ (τ ) : τ ∈ S}, κ ∈ S,
◦
◦
g ∗ (τ ) = sup{κ : τ − g(κ) : κ ∈ S}, τ ∈ S,
where g ∗ is the first Young transform of g. In our case we have
g ∗ (τ ) = max{g1∗ (τ ), g2∗ (τ ), g3∗ (τ )},
1 2
|τ | − g0 ,
4µ
1
g2∗ (τ ) = |τ |2 + ε0 : τ,
4
◦
1
g3∗ (τ ) = |τ |2 − ε0 : τ , τ ∈ S2 .
4
g1∗ (τ ) =
(2.3)
214
M. Fuchs, G. Seregin / A twodimensional variational model
We are not going to compute g ∗∗ , in place of this we discuss the so-called effective stressstrain relation
σ=
◦
∂g ∗∗
(κ), κ ∈ S,
∂κ
(2.4)
which has the following physical meaning: suppose that g ∗∗ (κ) < g(κ) with κ = ε(u(x))
for some point x ∈ Ω, u denoting a solution of problem QP. Then we say that at x ∈ Ω
a microstructure occurs which on a macro-level is described by the stress-strain relation
(2.4).
In Lemma 2.1 below we give explicit formulas for (2.4) in all possible cases which means
that the results of our computation heavily depend on the various choices for the parameters µ and g0 . Throughout the lemma we use latin numbers in brackets to indicate the
region of strains or stresses where microstructure can appear. For example, (I, III) means
that we have microstructure generated by the first and third phase.
Lemma 2.1. Let σ =
(a)
∂g ∗∗
(ε), ε
∂ε
◦
∈ S.
Suppose that
µ > 1.
(2.5)
Then we have

ε − ε0



σ = 2 ε + ε0


 ε − ε0 :ε ε
if
if
|ε0 |2 0
if
ε:ε0
|ε0 |2
ε:ε0
|ε0 |2
|ε:ε0 |
|ε0 |2
> 1
(II)
< −1 (III)
≤ 1
(2.6)
(II, III)
for g0 > 0,
σ=2
for g0 = 0,

ε − ε0




 ε + ε0

ε−




0
ε:ε0
ε
|ε0 |2 0
if
if
if
ε:ε0
|ε0 |2
ε:ε0
|ε0 |2
|ε:ε0 |
|ε0 |2
> 1
(II)
< −1
(III)
≤ 1, ε 6= αε0 (II, III)
if ε = αε0 , |α| ≤ 1
(I, II, III)
(2.7)
M. Fuchs, G. Seregin / A twodimensional variational model
σ=2








































ε0
|
aµ
ε0
|
aµ
µε
if |ε +
ε − ε0
if
ε:ε0
|ε0 |2
> 1, |ε +
ε + ε0
if
ε:ε0
|ε0 |2
< −1, |ε −
ε0
|
aµ
if
|ε:ε0 |
|ε0 |2
¬
¬
≤ 1, ¬ε −
¬
ε:ε0
ε
|ε0 |2 0 ¬
ε−
ε:ε0
ε
|ε0 |2 0
ε0
ε+ aµ
1 |ε+ ε0 |
aµ
ε0
a
< R2 , |ε −



R2 ≤ |ε +
ε0
|
aµ
(I)
> R1
(II)
> R1
¬
ε0
|
aµ
< R2
>
215
(III)
q
−g0
a
(II, III)
≤ R1 ,
−
if
R
¬
¬
q


¬
ε:ε0
ε:ε0
a ¬




1 − a + a |ε0 |2 > − g0 ¬ε − |ε0 |2 ε0 ¬








ε0


| ≤ R1 ,
R2 ≤ |ε − aµ




ε0

ε− aµ

ε

R1 |ε− ε0 | + a0
if

¬
¬
q

aµ


¬
ε:ε0
ε:ε0
a ¬




1 − a − a |ε0 |2 > − g0 ¬ε − |ε0 |2 ε0 ¬




¬
¬


¬
¬

ε:ε0


≤ − ga0 ,
ε
−
ε

¬

ε:ε0

|ε0 |2 0 ¬


ε
p
ε−
2 0


− ga0 ¬¬ |εε:ε0 |0 ¬¬ if

¬
¬

q
¬ε− |ε |2 ε0 ¬


¬
¬
|ε:ε0 |


ε:ε
a
0
0


1 − a + a |ε0 |2 ≤ − g0 ¬ε − |ε0 |2 ε0 ¬
(I, II)
(I, III)
(I, II, III)
(2.8)
for g0 < 0, where
1
a = 1 − , R1 =
µ
(b)
r
1
|ε0 |2 g0
− , R2 = R1 .
2
a
a
µ
(2.9)
Suppose that
µ=1
(2.10)
Then (2.6) for g0 > 0 and (2.7) for g0 = 0 are valid. Further we have

ε









ε − ε0





σ = 2 ε + ε0





ε:ε0


 ε − |ε0 |2 ε0 −





 ε − ε:ε0 ε +
|ε0 |2 0
(c)
if |ε : ε0 | < − g20
(I)
if ε : ε0 > − g20 + |ε0 |2
(II)
if ε : ε0 <
g0
2
− |ε0 |2
(III)
g0
ε
4|ε0 |2 0
if − g20 ≤ ε : ε0 ≤ − g20 + |ε0 |2 (I, II)
g0
ε
4|ε0 |2 0
if
g0
2
− |ε0 |2 ≤ ε : ε0 ≤
g0
2
(2.11)
(I, III)
for the case g0 < 0.
Suppose that
0 < µ < 1.
(2.12)
216
M. Fuchs, G. Seregin / A twodimensional variational model
Then we get
σ=2






































µε
if |ε +
ε0
|
bµ
> R4 , |ε −
ε − ε0
if |ε −
ε0
|
bµ
< R3 ,
ε:ε0
|ε0 |2
>1
(II)
ε + ε0
if |ε +
ε0
|
bµ
< R3 ,
ε:ε0
|ε0 |2
< −1
(III)
ε−
ε:ε0
ε
|ε0 |2 0
ε−
if
|ε:ε0 |
|ε0 |2
¬
¬
≤ 1, ¬ε −
ε0
|
bµ
¬
¬
ε:ε0
ε
|ε0 |2 0 ¬
> R4
<
(I)
p g0
(II, III)
b

ε0

R3 ≤ |ε − µb | ≤ R4 ,
if ¬
¬
°
±

¬¬ε − ε:ε0 ε ¬¬ > p g0 b + 1 − b ε:ε0
|ε0 |2 0
b
|ε0 |2
ε0
ε0
R3 |ε− µb
ε0 +
b
|

µb










ε0




R3 ≤ |ε + bµ | ≤ R4 ,
ε0

ε+

ε0


R3 |ε+ µb
if ¬
ε0 −
¬

°
±
b
|

µb


¬¬ε − ε:ε0 ε ¬¬ > p g0 b + 1 + b ε:ε0


|ε0 |2 0
b
|ε0 |2




p
p g0


g0
ε:ε0
1


≤
|ε
−
ε
|
≤
,

0
2
ε:ε

b
|ε0 |
µ
b

p g0 ε− |ε0 |02 ε0


¬
¬

if

b ¬¬ε− ε:ε0 ε0 ¬¬



|ε0 |2
|ε − ε:ε0 ε | ≤ p g0 (b + 1 − b |ε:ε0 | )
|ε0 |2 0
b
|ε0 |2
for g0 > 0, where b =
σ=2
1
µ
− 1, R3 =

µε








ε − ε0









ε + ε0




ε0

ε− bµ



R
3 |ε− ε0 | +


bµ




ε0

ε+ bµ



R
3 |ε+ ε0 | −


bµ



0
ε0
b
ε0
b
|ε0 |2
b2
+
g0
,
b
(I, II)
(I, III)
(I, II, III)
(2.13)
R4 = µ1 R3 ,
if |ε −
ε0
|
bµ
> R4 , |ε +
ε0
|
bµ
if |ε −
ε0
|
bµ
< R3
(II)
if |ε +
ε0
|
bµ
< R3
(III)
> R4 (I)
(
ε0
R3 ≤ |ε − bµ
| ≤ R4 ,
if
ε 6= γε0 , 0 ≤ γ ≤ 1
(
ε0
R3 ≤ |ε + bµ
| ≤ R4 ,
if
ε 6= γε0 , −1 ≤ γ ≤ 0
if ε = γε0 , |γ| ≤ 1
(2.14)
(I, II)
(I, III)
(I, II, III)
for g0 = 0,
σ = 2µε (I) for g0 < 0 and
|ε0 |2 g0
+
< 0,
b2
b

ε0
(I)
ε if ε 6= ± bµ


ε0
(I, II)
σ = 2µ ε if ε = bµ


ε0
ε if ε = − bµ
(I, III)
(2.15)
(2.16)
M. Fuchs, G. Seregin / A twodimensional variational model
for g0 < 0 and
σ=2
|ε0 |2
|b|2
+
g0
b
217
= 0,

µε









ε − ε0








 ε + ε0
ε0
|
bµ
if |ε −
ε0
|
bµ
> R4 , |ε +
if |ε −
ε0
|
bµ
< R3
(II)
if |ε +
ε0
|
bµ
< R3
(III)
> R4 (I)
(2.17)


ε:ε

ε− 02 ε0


¬ +
¬ |ε0 |
R

3
¬
¬
ε:ε0

ε0 ¬
¬ε−

2
|ε0 |





ε :ε

ε+ 0 2 ε0


¬ |ε0 |

R
 3 ¬ ε0 :ε ¬¬ −
¬ε+ |ε
2 ε0 ¬
ε0
b
if R3 ≤ |ε −
ε0
|
bµ
≤ R4
(I, II)
ε0
b
if R3 ≤ |ε +
ε0
|
bµ
≤ R4
(I, III)
0|
for g0 < 0 and
|ε0 |2
b2
+
g0
b
> 0.
Next we are going to state our results concerning the regularity properties of solutions
to problem QP. Partially they could be obtained at least in a qualitative sense by an
adoption of the techniques developed in [3]. The approach presented here is based on
duality theory which already has been applied to problems of phase transition in the
papers [20]–[22] and which also turned out useful in the context of plasticity theory (see,
e.g.[10]). In our opinion duality methods are quite effective since they give more precise
regularity results, moreover, it is possible to formulate integral conditions whether a
microstructure occurs at some point of the body or not. The dual variational problem P ∗
is introduced as follows:
Problem P ∗ :
Find a tensor σ ∈ Qf such that
R(σ) = sup{R(τ ) : τ ∈ Qf }.
Here R denotes the functional
R(τ ) =
Z
(ε(u0 ) : τ − f · u0 − g ∗ (τ ))dx
Ω
defined for tensors τ from the set


Z


◦
◦
Qf = τ ∈ L2 (Ω; S) : (τ : ε(v) − f · v)dx = 0 for all v ∈ J 12 (Ω) .


Ω
We recall (see [9]) that P ∗ has a unique solution σ; if u denotes a solution of QP, then
we have the duality relation
σ(x) =
∂g ∗∗
(ε(u)(x)) for almost all x ∈ Ω
∂κ
(2.18)
as well as the equation
QI(u) = R(σ).
(2.19)
218
M. Fuchs, G. Seregin / A twodimensional variational model
1
Theorem 2.2. Suppose in addition to (2.1) that f ∈ W2,loc
(Ω; R2 ).
Then the solution σ of problem P ∗ has weak derivatives in the space L2loc , i.e. we have
◦
1
(Ω; S).
σ ∈ W2,loc
(2.20)
Remark 2.3. The statement of Theorem 2.2 holds in a quite more general setting which
means that the proof just uses convexity of g ∗∗ together with the boundedness of the
second derivatives, in particular, no condition like strict convexity is needed to carry out
the proof.
As before we let σ denote the unique maximizer of the functional R and consider the set
of all Lebesgue points of σ, i.e. the set
Ω0 =
º
x ∈ Ω : lim(σ)x,R exists
R↓0
»
.
Here we use the symbols
(f )x,R
Z
= − f dy =
BR
(x)
1
|BR |
Z
BR
f dy
(x)
to denote the mean value of a function f w.r.t. the disc BR (x) with radius R and center
at x ∈ R2 . For the particular case x = 0 we just write BR and (f )R in place of BR (0) and
(f )0,R .
Next we let
A = {1, 2, 3}, A(τ ) = {i ∈ A : g ∗ (τ ) = gi∗ (τ )},
a(σ) = {x ∈ Ω0 : card A(σ(x)) = 1}.
The physical meaning of the set a(σ) is that it can be seen as the union of single phases
and that at the points of a(σ) no microstructure occurs. Our main regularity result reads
as follows:
Theorem 2.4. Suppose that all the conditions of Theorem 2.2 hold. If in addition f
1
2
belongs to the space L∞
loc ∩ W2,loc (Ω; R ), then the set a(σ) is open and σ is Hölder continuous on a(σ) for any exponent 0 < α < 1. Moreover, card A(σ(x)) > 1 for almost all
x ∈ Ω − a(σ).
3.
Proof of Lemma 2.1
We are not going to prove Lemma 2.1 for all possible cases, instead of this we restrict
ourselves to a representative situation, for example g0 < 0 together with µ > 1, i.e. we
are going to prove relation (2.8).
In order to compute
∂g ∗∗
(ε)
∂ε
we let
◦
S 3 τ → Fε (τ ) = g ∗ (τ ) − τ : ε
M. Fuchs, G. Seregin / A twodimensional variational model
219
and observe the relation
σ=
∂g ∗∗
(ε)
∂ε
⇐⇒
0 ∈ ∂Fε (σ),
(3.1)
◦
where ∂Fε (σ) is the subdifferential of the function Fε at σ ∈ S.
As in Section 2 we let
◦
A(τ ) = {i ∈ A : gi∗ (τ ) = g ∗ (τ )},
A = {1, 2, 3},
τ ∈ S.
(3.2)
◦
Then, for any τ ∈ S, we have
∂Fε (τ ) =
X
λi ∂gi∗ (τ ) − ε
(3.3)
i∈A(τ )
with suitable numbers λi ≥ 0,
i ∈ A(τ ), satisfying
X
λi = 1.
(3.4)
i∈A(τ )
It is easy to see that
∂g1∗ (τ )
=
º
1
τ
2µ
»
,
∂g2∗ (τ )
=
º
º
»
»
1
1
∗
τ + ε0 , ∂g3 (τ ) =
τ − ε0 .
2
2
(3.5)
Finally, we introduce the functions
◦
Fij (τ ) = gi∗ (τ ) − gj∗ (τ ), i, j = 1, 2, 3, τ ∈ S,
(3.6)
1 2
1
|τ | − g0 − |τ |2 − τ : ε0 =
4µ
4
2ε0 2
a
(4R12 − |τ +
| ),
4
a
2ε0 2
a
F31 (τ ) = (|τ −
| − 4R12 ),
4
a
F23 (τ ) = 2τ : ε0
(3.7)
and observe
F12 (τ ) =
◦
for any τ ∈ S, the quantities a and R1 being defined in (2.9).
The following cases can occur (observe F12 + F23 + F31 = 0):
F12 (τ )
F23 (τ )
F31 (τ )
F23 (τ )
F23 (τ )
F23 (τ )
F12 (τ )
>
>
>
=
>
<
=
0,
0,
0,
0,
0,
0,
0,
F31 (τ )
F12 (τ )
F23 (τ )
F12 (τ )
F12 (τ )
F31 (τ )
F31 (τ )
<
<
<
<
=
=
=
0,
0,
0,
0 (=⇒ F31 (τ ) > 0),
0,
0,
0 (=⇒ F23 (τ ) = 0).
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
220
M. Fuchs, G. Seregin / A twodimensional variational model
∗∗
Next we recall that σ = ∂g∂ε (ε) and suppose at first that (3.8) is valid for the tensor σ.
From (3.6) it follows that A(σ) = {1} and therefore, by (3.3)–(3.5), we get σ = 2µε, hence
the first line in (2.8) is proved. The next cases (3.9) and (3.10) are treated in the same
way.
Now let us consider (3.11), i.e. F23 (σ) = 0 together with F12 (σ) < 0.
Then
A(σ) = {2, 3}
and
(recall (3.3), (3.4))
σ=
ε − (λ2 − λ3 )ε0
λ2
+ λ23
2
with λ2 , λ3 ≥ 0, λ2 + λ3 = 1. In order to calculate λ2 , λ3 we observe that
implies
λ2 − λ3 =
F23 (σ) = 0
ε : εo
,
|ε0 |2
hence
ε0 : ε
ε0 ),
|ε0 |2
1
ε : ε0
λ2 = (1 +
),
2
|ε0 |2
1
ε : ε0
λ3 = (1 −
).
2
|ε0 |2
σ = 2(ε −
(3.15)
Since λ2 , λ3 ≥ 0, we see that ε must satisfy the restriction
|ε : ε0 |
≤ 1.
|ε|2
(3.16)
Recalling F12 (σ) < 0 and inserting (3.15), we get
"
¬
¬2 #
² ²
³³
ε : ε0 ¬¬
ε : ε0
g0 ¬¬
< 0,
F12 2 ε −
ε0
= −a
+ ε−
ε0
|ε0 |2
a ¬
|ε0 |2 ¬
and this together with (3.15), (3.16) implies the fourth line in (2.8). Suppose next that σ
satisfies (3.12). Then A(σ) = {1, 2}, hence
σ=
ε − λ2 ε 0
, λ1 , λ2 ≥ 0, λ1 + λ2 = 1.
λ1
+ λ22
2µ
M. Fuchs, G. Seregin / A twodimensional variational model
Rewriting
F12 (σ) = 0
F12
221
as
À
ε − λ2 ε 0
λ1
+ λ22
2µ
!
²
³
ε − λ2 ε 0
= F12 2
1 − a + aλ2
"
¬
¬2 #
¬
ε
−
λ
ε
ε
2 0
0¬
= a R12 − ¬¬
+ ¬¬
1 − a + aλ2
a
#
"
ε0 2
|ε + aµ
|
2
= a R1 −
(1 − a + aλ2 )2
=0
we obtain

λ2
σ
ε0
= a1 (− µ1 + R11 |ε + aµ
|)
h ε+ ε0
i
ε0
= 2 R1 |ε+ aµ
,
ε0 −
a
|
(3.17)
aµ
and 0 ≤ λ2 ≤ 1 implies the inequality
¬
¬
¬
¬
ε
0
¬ ≤ R1 .
R2 ≤ ¬¬ε +
aµ ¬
(3.18)
In addition we know
1
F23 (σ) > 0 ⇐⇒ λ2 =
a
²
1
1
− +
µ R1
¬
¬³
¬
¬
ε
0
¬ε +
¬ < ε : ε0 ,
¬
aµ ¬
|ε0 |2
hence
ε : ε0
1−a+a
>
|ε0 |2
r
a
−
g0
¬
¬
¬
¬
ε
:
ε
0
¬ε −
¬,
ε
0
¬
|ε0 |2 ¬
and this together with (3.17) and (3.18) completes the fifth line in formula (2.8). Case
(3.13) is treated in the same way, and it remains to discuss (3.14). Now A(σ) = {1, 2, 3}
and
ε − (λ2 − λ3 )ε0
σ = λ1 λ2 λ3 , λ1 , λ2 , λ3 ≥ 0, λ1 + λ2 + λ3 = 1.
+ 2 + 2
2µ
F23 (σ) = 0 gives
ε − |εε:ε0 |02 ε0
ε : ε0
λ2 − λ3 =
,σ = 2
.
|ε0 |2
1 − a + a(λ2 + λ3 )
From
F12 (σ) = F23 (σ) = 0
we deduce
1
λ2 + λ3 =
a
²r
a
−
g0
|σ|2 = − ga0 , so that
¬
¬
³
¬
¬ 1
ε
:
ε
0
¬ε −
ε0 ¬ −
.
¬
|ε0 |2 ¬ µ
222
M. Fuchs, G. Seregin / A twodimensional variational model
This implies
¬
¬
¬
¬
¬ ε − ε : ε0 ε0 ¬ −
¬
|ε0 |2 ¬
¬
¬
²r
ε : ε0 ¬¬
1
a ¬¬
2λ3 =
− ¬ε −
ε0 −
a
g0
|ε0 |2 ¬
r
ε:ε0
g0 ε − |ε0 |2 ε0
¬.
σ = 2 − ¬¬
a ¬ε − ε:ε0 ε ¬¬
|ε0 |2 0
1
2λ2 =
a
²r
a
−
g0
1
µ
³
+
ε : ε0
,
|ε0 |2
1
µ
³
−
ε : ε0
,
|ε0 |2
Taking into account the restrictions λ2 ≥ 0, λ3 ≥ 0, λ2 + λ3 ≤ 1, the last line of (2.8) is
established which completes the proof of Lemma 2.1.
2
4.
Proof of Theorem 2.2
◦
∗∗
From Lemma 2.1 we deduce that ∂g∂ε is Lipschitz continuous on S, hence there exists
c1 ≥ 0 such that
¬ 2 ∗∗ ¬
¬D g (ε)¬ ≤ c1
(4.1)
◦
for all ε ∈ S. Quoting [15] we find a pressure function p ∈ L2 (Ω) with the property
Z
Z
Z
σ : ε(v) dx = p div v dx + f · v dx
(4.2)
Ω
Ω
Ω
◦
being valid for all v ∈ W 12 (Ω; R2 ). Consider a disc BR (x0 ) with compact closure in Ω and
choose ϕ ∈ C0∞ (Ω) according to
ϕ = 0 outside of Br (x0 ),
ϕ=1
on Bq (x0 ),
c2
|∇ϕ| ≤
, 0 ≤ ϕ ≤ 1 in Ω.
r−q
(4.3)
¢
£
Here q < r denote arbitrary numbers in the interval R2 , R . For h ∈ R2 with sufficiently
small norm we have ±h + BR (x0 ) ⊂ Ω. For functions w let us define ∆h w(x) =
w(x + h) − w(x). Then we obtain from (4.2)
Z
Z
2
∆h f · ϕ2 ∆h u dx
∆h σ : ε(ϕ ∆h u) dx =
BR (x0 )
BR (x0 )
+
Z
BR (x0 )
(4.4)
2
g
∆
h p div(ϕ ∆h u) dx,
where
◦
u(x) = u(x) − A(x − x0 ) − u0 , A ∈ M, u0 ∈ R2 ,
g
∆
h p = ∆h p − (∆h p)x0 ,r .
M. Fuchs, G. Seregin / A twodimensional variational model
223
Next we introduce the parameter dependent bilinear form
L(x) =
Z1
D2 g ∗∗ (ε(u)(x) + Θε(∆h u)(x)) dΘ, x ∈ Ω.
0
We have
◦
(L(x){) : { ≥ 0, { ∈ S, x ∈ Ω, |L(x)| ≤ c1 .
(4.5)
Note that L(x) is defined only at almost all points x of Ω. Observing ∆h σ = L ε(∆h u)
we find that
|∆h σ|2 = L ε(∆h u) : ∆h σ ≤
1
1
(ε(∆h u) : L ε(∆h u)) 2 (∆h σ : L∆h σ) 2 ≤
1 √
(ε(∆h u) : ∆h σ) 2 c1 |∆h σ| , i.e.
|∆h σ|2 ≤ c1 ∆h σ : ε(∆h u).
Using this estimate in equation (4.4) and recalling (4.3) we get
Z
BR (x0 )



Z
1
ϕ |∆h σ| dx ≤ c3
2

 (r − q)
2
2
|∆h u|2 dx
BR (x0 )
+ (r − q)2
Z
|∆h f |2 dx +
BR (x0 )
c4




Z
BR (x0 )



2
g
∆h p∇ϕ · ∆h u dx ≤



Z
Z
 21
1 
1

2
g
|∆
|∆h u|2 dx +

h p| dx
2

(r − q)
r−q


Br (x0 )
BR (x0 )
·
 12



Z
Z



2
2
2
|∆h f | dx . (4.6)
|∆h u| dx + R




BR (x0 )
BR (x0 )

◦
By results of [15] there exists w ∈ W 12 (Br (x0 ); R2 ) such that
g
div w = ∆
h p in Br (x0 ),
(4.7)
g
k∇wkL2 (Br (x0 )) ≤ c5 k∆
h pkL2 (Br (x0 )) ,
the constant c5 being independent of x0 and r. (4.2) together with (4.7) implies
Z
Z
Z
2
g
∆h σ : ε(w) dx =
∆h f · w dx +
|∆
h p| dx
Br (x0 )
Br (x0 )
Br (x0 )
224
M. Fuchs, G. Seregin / A twodimensional variational model
so that

Z
Br (x0 )

2
g
|∆
h p| dx ≤ c6 
Z
|∆h σ|2 dx + R2
Br (x0 )

Z
BR (x0 )

|∆h f |2 dx .
(4.8)
Inserting (4.8) into (4.6) and using Young’s inequality we get
Z
|∆h σ|2 dx ≤
Bq (x0 )
1
2
Z
Br (x0 )


|∆h σ|2 dx + c7 
1
(r − q)2
Z
|∆h u|2 dx + R2

Z
BR (x0 )
BR (x0 )

|∆h f |2 dx
¢R
£
,
R
. As in [11] this implies
2
for all q < r in

Z
B R (x0 )
2
 1
|∆h σ|2 dx ≤ c8  2
R
Z
|∆h u|2 dx + R2

Z
BR (x0 )
BR (x0 )

|∆h f |2 dx
for any BR (x0 ) ⊂ Ω. If we replace h by λe for some unit vector e ∈ R2 and divide the
above inequality by |λ|, we see that the right-hand side has a limit as λ → 0 which can
be bounded by

 1
c8  2
R
Z
|ε(u) − {|2 dx + R2
BR (x0 )
Z
BR (x0 )


|∇f |2 dx
◦
where { is some matrix in S. (To prove this choose A and u0 in an appropriate way
and use Korn’s inequality.) Putting together our results we have shown that σ has weak
derivatives in L2loc (Ω), moreover, we have the estimate
Z
B R (x0 )
2

 1
|∇σ|2 dx ≤ c8  2
R
Z
BR (x0 )
|ε(u) − {|2 dx + R2
Z
BR (x0 )


|∇f |2 dx
(4.9)
◦
being valid for any disc BR (x0 ) with compact closure in Ω and for any matrix { ∈ S.
This completes the proof of Theorem 2.2.
2
M. Fuchs, G. Seregin / A twodimensional variational model
5.
225
Local estimates of Caccioppoli-type
We are going to consider the unique solution σ of the dual problem P ∗ and introduce the
following quantities related to this tensor:
∆12
x
∆12
x,ρ
¬
¬
¬
2ε0 ¬¬
¬
= 2R1 − ¬σ(x) +
,
a ¬
¬
¬
¬
2ε0 ¬¬
¬
= 2R1 − ¬(σ)x,ρ +
,
a ¬
ε0
ε0
, ∆23
,
x,ρ = (σ)x,ρ :
|ε0 |
|ε0 |
¬
¬
¬
2ε0 ¬¬
¬
= ¬σ(x) −
− 2R1 ,
a ¬
¬
¬
¬
2ε0 ¬¬
¬
− 2R1 .
= ¬(σ)x,ρ −
a ¬
∆23
x = σ(x) :
∆31
x
∆31
x,ρ
(5.1)
Here x is taken from the Lebesgue set Ω0 of σ and ρ > 0◦ is a radius such that Bρ (x) ⊂ Ω.
We further recall the definition of the functions Fij : S → R (see (3.6) and (3.7)) and
observe
> (<) 0,
Fij (σ(x)) > (<)0 ⇐⇒ ∆ij
x
(5.2)
ij
Fij ((σ)x,ρ ) > (<)0 ⇐⇒ ∆x,ρ > (<) 0
for
i = 1, j = 2, i = 2, j = 3 and i = 3, j = 1.
Lemma 5.1. Suppose that all the hypotheses of Theorem 2.4 are valid. Then we have the
following statements:
(i)
∆12
x0 ,R > 0
If
Z
|∇σ|2 dx ≤ c1
and
h ∆31
x0 ,R < 0,
then
¡
−2
−2
·
+ |∆31
1 + |∆12
x0 ,R |
x0 ,R |
B R (x0 )
2
1
· 2
R
Z
2
|σ − (σ)x0 ,R | dx + R
BR (x0 )
(ii)
∆23
x0 ,R > 0
If
Z
2
|∇σ| dx ≤ c2
and
h 2
Z
i
|∇f |2 dx . (5.3)
BR (x0 )
∆12
x0 ,R < 0,
then
¡
−2
−2
·
+ |∆12
1 + |∆23
x0 ,R |
x0 ,R |
B R (x0 )
2
1
· 2
R
Z
BR (x0 )
2
|σ − (σ)x0 ,R | dx + R
2
Z
BR (x0 )
i
|∇f | dx . (5.4)
2
226
M. Fuchs, G. Seregin / A twodimensional variational model
∆31
x0 ,R > 0
(iii) If
Z
∆23
x0 ,R < 0,
and
|∇σ|2 dx ≤ c3
h then
¡
−2
−2
·
+ |∆23
1 + |∆31
x0 ,R |
x0 ,R |
B R (x0 )
2
Z
1
· 2
R
2
|σ − (σ)x0 ,R | dx + R
BR (x0 )
2
Z
i
|∇f |2 dx . (5.5)
BR (x0 )
The estimates are valid for any x0 ∈ Ω0 and any R > 0 s.t. BR (x0 ) ⊂⊂ Ω. Finally, the
positive constants c1 , c2 , c3 do not depend on x0 and R.
Proof. Our arguments closely follow the lines of the papers [20]–[22].
We let
©
ω ij (x0 , R) = {x ∈ BR (x0 ) : ∆ij
x ≤ 0 ,
©
Ωij (x0 , R) = {x ∈ BR (x0 ) : ∆ij
x ≥ 0 ,
©
ij
ω−
(x0 , R) = {x ∈ BR (x0 ) : ∆ij
x < 0 ,
(5.6)
©
=
0
,
ω0ij (x0 , R) = {x ∈ BR (x0 ) : ∆ij
x
©
ij
>
0
ω+
(x0 , R) = {x ∈ BR (x0 ) : ∆ij
x
for i = 1 and j = 2, i = 2 and
R j = 3, i = 3 2and j = 1 and recall formula (4.9) in which we
have to estimate the term
|ε(u) − {| dx. So let us first consider case (i) of Lemma
BR (x0 )
5.1. In order to prove (5.3) we define
1
(σ)x0 ,R
2µ
{=
(5.7)
and observe that (5.1) together with (5.2) implies the bound
²
³
|ε0 |
|(σ)x0 ,R | ≤ 2 R1 +
a
(5.8)
Next we use the decomposition
Z
BR (x0 )
1
|ε(u) − {| dx =
(2µ)2
2
Z
|σ − (σ)x0 ,R |2 dx
12 (x ,R)∩ω 31 (x ,R)
ω+
0
0
−
+
Z
ω 12 (x0 ,R)∪Ω31 (x0 ,R)
¬
¬2
¬
¬
1
¬ε(u) −
¬ dx. (5.9)
(σ)
x
,R
0
¬
¬
2µ
M. Fuchs, G. Seregin / A twodimensional variational model
227
The second integral on the right-hand side of (5.9) can be splitted in the following way
(recall that F12 + F23 + F31 = 0):
Z
I=
ω 12 ∪Ω31
¬
¬2
¬
¬
¬ε(u) − 1 (σ)x0 ,R ¬ dx = I2 + I3 + . . . + I7 ,
¬
¬
2µ
(5.10)
where
I2 =
Z
12 ∩ω 23
ω−
+
I3 =
Z
31 ∩ω 23
ω+
−
¬
¬2
¬
¬
1
¬ε(u) −
(σ)x0 ,R ¬¬ dx,
¬
2µ
¬
¬2
¬
¬
1
¬ε(u) −
¬ dx,
(σ)
x
,R
0
¬
¬
2µ
¬
¬2
¬
¬
1
¬ε(u) −
¬ dx,
(σ)
x
,R
0
¬
¬
2µ
Z
I4 =
(ω−12 ∪ω+31 )∩ω023
¬2
Z ¬
¬
¬
1
¬ε(u) −
I5 =
(σ)x0 ,R ¬¬ dx,
¬
2µ
23
ω012 ∩ω+
I6 =
Z
23
ω031 ∩ω−
I7 =
¬
¬2
¬
¬
1
¬ε(u) −
(σ)x0 ,R ¬¬ dx,
¬
2µ
Z
ω031 ∩ω012 ∩ω023
¬
¬2
¬
¬
1
¬ε(u) −
¬ dx .
(σ)
x
,R
0
¬
¬
2µ
Here and in what follows we will not explicitly indicate x0 and R in the domain of
definition. The numeration of the integrals I2 , · · · , I7 is in correspondence with the lines
of formula (2.8)where the relation between σ and ε(u) for the various cases is given. Using
this together with (5.8) we find
Z
I2 =
12 ∩ω 23
ω−
+

1
≤ 2
4
≤
1
2
¬
¬2
¬1
¬
1
¬ σ + ε0 − (σ)x0 ,R ¬ dx
¬2
¬
2µ
Z
BR (x0 )
Z
BR (x0 )

³
²
¬
¬ 12
1
23 ¬
∩ ω+
|σ − (σ)x0 ,R |2 dx + |ε0 |2 + a |(σ)x0 ,R |2 ¬ω−

2
¬ ¬ ¬ ¬¡
|σ − (σ)x0 ,R |2 dx + c4 ¬ω 12 ¬ + ¬Ω31 ¬ ,
228
M. Fuchs, G. Seregin / A twodimensional variational model
1
I3 ≤
2
Z
¬ ¬ ¬ ¬¡
|σ − (σ)x0 ,R |2 dx + c5 ¬ω 12 ¬ + ¬Ω31 ¬ ,
BR (x0 )
Z
I4 =
12 ∪ω 31 )∩ω 23
(ω−
0
+
1
≤
2
Z
¬
¬2
¬1
¬
ε
:
ε
1
0
¬ σ+
¬ dx
ε
−
(σ)
0
x
,R
0
¬2
¬
|ε0 |2
2µ
¡
|σ − (σ)x0 ,R |2 dx + c6 |ω 12 | + |Ω31 | ,
BR (x0 )
I5 ≤ 2
´
µ
1
2
2
|ε(u)| +
|(σ)x0 ,R | dx
2µ
Z
23
ω012 ∩ω+
"²
³2
²
³2 #
¬
¬ 12
1
|ε0 |
|ε0 |
23 ¬
¬ω0 ∩ ω+
≤ 2 R1 +
+ 2 4 R1 +
aµ
4µ
a
¬ 12 ¬ ¬ 31 ¬¡
≤ c7 ¬ ω ¬ + ¬ Ω ¬ ,
¬ ¬ ¬ ¬¡
I6 ≤ c8 ¬ω 12 ¬ + ¬Ω31 ¬ ,
I7 ≤ 2
´
µ
1
2
2
|ε(u)| + 2 |(σ)x0 ,R | dx
4µ
Z
ω012 ∩ω023 ∩ω031
"
³2 #
²
° g
±
12
¡
1
|ε0 |
0
≤ 2 − + |ε0 |2 + 2 R1 +
|ω | + |Ω31 |
a
µ
a
and therefore


I ≤ c9 
Z
BR (x0 )

¬ ¬ ¬ ¬
|σ − (σ)x0 ,R |2 dx + ¬ω 12 ¬ + ¬Ω31 ¬ .
It remains to discuss the measures of the sets ω 12 (x0 , R) and Ω31 (x0 , R).
(5.11)
M. Fuchs, G. Seregin / A twodimensional variational model
229
We have
¬−1
¬ 12
¬ ¬
¬
¬ω (x0 , R)¬ = ¬∆12
x ,R
Z
¬−1
¬
¬
≤ ¬∆12
x ,R
Z
¬−1
¬
¬
≤ ¬∆12
x ,R
Z
0
∆12
x0 ,R dx
ω 12 (x0 ,R)
0
¡
12
dx
−
∆
∆12
x
x0 ,R
ω 12 (x0 ,R)
0
|σ − (σ)x0 ,R | dx
ω 12 (x0 ,R)
1
2
≤

|ω 12 (x0 , R)| 
¬
¬
¬∆12 ¬ 
x0 ,R
 12
Z
BR (x0 )
¬−1
¬ 31
¬ ¬
¬
¬Ω (x0 , R)¬ = ¬∆31
x ,R
Z
¬−1
¬
¬
≤ ¬∆31
x ,R
Z
¬−1
¬
¬
≤ ¬∆31
x ,R
Z
0

|σ − (σ)x0 ,R |2 dx ,
¡
−∆31
x0 ,R dx
Ω31 (x0 ,R)
0
¡
31
∆31
x − ∆x0 ,R dx
Ω31 (x0 ,R)
0
|σ − (σ)x0 ,R | dx
Ω31 (x0 ,R)
1
2
≤

|Ω31 (x0 , R)| 
¬
¬
¬∆31 ¬ 
x0 ,R
Z
BR (x0 )
 21

|σ − (σ)x0 ,R |2 dx .
From this together with (4.9), (5.9) and (5.11) part (i) of Lemma 5.1 will follow.
Now we are going to prove inequality (5.4). In this case we let
1
{ = (σ)x0 ,R + ε0
2
and obtain
Z
1
|ε(u) − {| dx =
(2µ)2
2
Z
|σ − (σ)x0 ,R |2 dx
23 ∩ω 12
ω+
−
BR (x0 )
(5.12)
+
Z
ω 23 ∪Ω12
¬
¬2
¬
¬
¬ε(u) − 1 (σ)x0 ,R − ε0 ¬ dx.
¬
¬
2
As in the previous case we split
0
I =
Z
ω 23 ∪Ω12
¬
¬2
¬
¬
1
¬ε(u) − (σ)x0 ,R − ε0 ¬ dx
¬
¬
2
= I10 + I30 + I40 + I50 + I60 + I70 ,
(5.13)
230
M. Fuchs, G. Seregin / A twodimensional variational model
where
Z
I10 =
12 ∩ω 31
ω+
−
I30
Z
=
31 ∩ω 23
ω+
−
I40
¬
¬2
¬
¬
1
¬ε(u) − (σ)x0 ,R − ε0 ¬ dx,
¬
¬
2
¬
¬2
¬
¬
¬ε(u) − 1 (σ)x0 ,R − ε0 ¬ dx,
¬
¬
2
Z
=
(ω−12 ∪ω+31 )∩ω023
I50
Z
=
23
ω012 ∩ω+
I60
Z
=
23
ω031 ∩ω−
¬
¬2
¬
¬
1
¬ε(u) − (σ)x0 ,R − ε0 ¬ dx,
¬
¬
2
¬
¬2
¬
¬
1
¬ε(u) − (σ)x0 ,R − ε0 ¬ dx,
¬
¬
2
Z
I70 =
¬
¬2
¬
¬
¬ε(u) − 1 (σ)x0 ,R − ε0 ¬ dx,
¬
¬
2
ω012 ∩ω023 ∩ω031
¬
¬2
¬
¬
1
¬ε(u) − (σ)x0 ,R − ε0 ¬ dx,
¬
¬
2
and the numeration of the integrals corresponds to the numeration of the lines in (2.8).
Proceeding as before we find
I10
¬
¬2
¬1
¬
1
¬ σ − (σ)x0 ,R − ε0 ¬ dx
¬ 2µ
¬
2
Z
=
12 ∩ω 31
ω+
−

Z
1
≤ 2
4
|σ − (σ)x0 ,R |2 dx +
BR (x0 )
On
31
12
∩ ω−
ω+
1
≤
2
12 ∩ω 31
ω+
−

¬
¬2
¬1
¬
¬ aσ − ε0 ¬ dx
.
¬2
¬
|σ| ≤ 2R1 + 2 |εa0 | , hence
we have
I10
Z
Z
BR (x0 )
¬ ¬ ¬ ¬¡
|σ − (σ)x0 ,R |2 dx + c10 ¬ω 23 ¬ + ¬Ω12 ¬ .
M. Fuchs, G. Seregin / A twodimensional variational model
231
Further we have
I30
Z
=
31 ∩ω 23
ω+
−
1
≤
2
Z
BR (x0 )
I40
¬
¬2
¬1
¬
¬ σ − ε0 − 1 (σ)x0 ,R − ε0 ¬ dx
¬2
¬
2
¬ ¬ ¬ ¬¡
|σ − (σ)x0 ,R |2 dx + 8|ε0 |2 ¬ω 23 ¬ + ¬Ω12 ¬ ,
¬
¬2
¬1
¬
¬ σ + ε : ε0 ε0 − 1 (σ)x0 ,R − ε0 ¬ dx
¬2
¬
2
|ε0 |
2
Z
=
(ω−12 ∪ω+31 )∩ω023
Z
¬ ¬ ¬ ¬¡
1
≤
|σ − (σ)x0 ,R |2 dx + 8|ε0 |2 ¬ω 23 ¬ + ¬Ω12 ¬ ,
2
BR (x0 )
I50
Z
=
23
ω012 ∩ω+
1
≤
2
Z
¬
¬2
¬
¬
1
1
¬ε(u) − σ + (σ − (σ)x0 ,R ) − ε0 ¬ dx
¬
¬
2
2
2
|σ − (σ)x0 ,R | dx + 2
≤
1
2
BR (x0 )
I60
1
≤
2
Z
BR (x0 )
I70
=
¡
|ε(u)|2 + |σ|2 + |ε0 |2 dx
23
ω012 ∩ω−
BR (x0 )
Z
Z
¬ ¬ ¬ ¬¡
|σ − (σ)x0 ,R |2 dx + c11 ¬ω 23 ¬ + ¬Ω12 ¬ ,
¬ ¬ ¬ ¬¡
|σ − (σ)x0 ,R |2 dx + c12 ¬ω 23 ¬ + ¬Ω12 ¬ ,
¬
¬2
¬
¬
¬ε(u) − 1 σ + 1 (σ − (σ)x0 ,R ) − ε0 ¬ dx,
¬
¬
2
2
Z
(ω012 ∩ω023 )∩ω031
Z
¬ ¬ ¬ ¬¡
1
≤
|σ − (σ)x0 ,R |2 dx + c13 ¬ω 23 ¬ + ¬Ω12 ¬ ,
2
BR (x0 )
and we arrive at (recall (5.13))


I 0 ≤ c14 
Z
BR (x0 )

¬ ¬ ¬ ¬
|σ − (σ)x0 ,R |2 dx + ¬ω 23 ¬ + ¬Ω12 ¬ .
(5.14)
232
M. Fuchs, G. Seregin / A twodimensional variational model
It is easy to show that the estimates
¬−2
¬ 23
¬ ¬
¬
¬ω (x0 , R)¬ ≤ ¬∆23
x ,R
Z
¬−2
¬ 12
¬ ¬
¬
¬Ω (x0 , R)¬ ≤ ¬∆12
x ,R
Z
0
|σ − (σ)x0 ,R |2 dx,
BR (x0 )
0
|σ − (σ)x0 ,R |2 dx
BR (x0 )
hold true, and (5.4) is a consequence of (4.9) and (5.12) - (5.14).
The proof of Lemma 5.1 (iii) is very close to the previous case, the necessary adjustments
are left to the reader. Altogether the claim of Lemma 5.1 is established with constants
c1 , c2 and c3 not depending on x0 and R.
6.
A decay estimate
We continue our discussion of the properties of the maximizer σ. As usual the behaviour
of σ is described in terms of the squared mean oscillation which is given by
 12
Z


Ψ(x0 , R) =  − |σ − (σ)x0 ,R |2 dx .

BR (x0 )
Recalling (5.1) we further let
¬ 12 ¬−p ¬¬ 12 ¬¬−p ¬ 31 ¬−p ¬¬ 31 ¬¬−p
Γ23 (x0 , R, p) = 1 + ¬∆x0 ,R ¬ + ¬∆x0 , R ¬ + ¬∆x0 ,R ¬ + ¬∆x0 , R ¬ ,
2
2
¬ 23 ¬−p ¬¬ 23 ¬¬−p ¬ 12 ¬−p ¬¬ 12 ¬¬−p
Γ31 (x0 , R, p) = 1 + ¬∆x0 ,R ¬ + ¬∆x0 , R ¬ + ¬∆x0 ,R ¬ + ¬∆x0 , R ¬ ,
2
2
¬ 31 ¬−p ¬¬ 31 ¬¬−p ¬ 23 ¬−p ¬¬ 23 ¬¬−p
Γ12 (x0 , R, p) = 1 + ¬∆x0 ,R ¬ + ¬∆x0 , R ¬ + ¬∆x0 ,R ¬ + ¬∆x0 , R ¬ ,
2
2
whenever these expressions make sense. The next result is an analogue of Lemma 7.3 in
[22] where a decay estimate is established for the case of two wells.
Lemma 6.1. Suppose that all the hypotheses of Theorem 2.4 are satisfied. Fix some
number p > 2 and consider a domain Ω0 with compact closure in Ω. Then, for any
x0 ∈ Ω0 and any 0 < R ≤ 12 R0 = 12 dist(∂Ω, Ω0 ), the following statements are true:
(i)
If
31
12
31
∆12
x0 ,R > 0, ∆x0 , R > 0, ∆x0 ,R < 0, ∆x0 , R < 0,
2
(6.1)
2
then
Ψ(x0 , ρ) ≤ c1
º´
µ
p
ρ
R
−1
+ Γ23 (x0 , R, p)Ψ 2 (x0 , R) ·
R
ρ
»
R
· Ψ(x0 , R) + Γ23 (x0 , R, p)R , 0 < ρ ≤ R. (6.2)
ρ
M. Fuchs, G. Seregin / A twodimensional variational model
(ii)
233
If
12
23
12
∆23
x0 ,R > 0, ∆x0 , R > 0, ∆x0 ,R < 0, ∆x0 , R < 0,
2
(6.3)
2
then
Ψ(x0 , ρ) ≤ c1
º´
µ
p
ρ
R
−1
+ Γ31 (x0 , R, p)Ψ 2 (x0 , R) ·
R
ρ
»
R
· Ψ(x0 , R) + Γ31 (x0 , R, p)R , 0 < ρ ≤ R. (6.4)
ρ
(iii) If
23
31
23
∆31
x0 ,R > 0, ∆x0 , R > 0, ∆x0 ,R < 0, ∆x0 , R < 0,
2
(6.5)
2
then
Ψ(x0 , ρ) ≤ c1
º´
µ
p
ρ
R
−1
+ Γ12 (x0 , R, p)Ψ 2 (x0 , R) ·
R
ρ
»
R
· Ψ(x0 , R) + Γ12 (x0 , R, p)R , 0 < ρ ≤ R. (6.6)
ρ
Here c1 denotes a positive constant independent of x0 , ρ and R.
Proof. Let (6.1) hold and consider the following Neumann boundary value problem:
°
±
°
◦±
to find uR ∈ J21 B R (x0 ) and σR ∈ L2 B R (x0 ); S such that
2
2
σR = 2 ε(uR ) in B R (x0 ),
2
Z
(σR − σ) : ε(w) dx = 0
for any
°
±
w ∈ J21 B R (x0 ) .
(6.7)
2
B R (x
2
0)
Equations (6.7) determine σR in an unique way, whereas uR is fixed up to an incompressible rigid motion. From [22] we get the estimate
° Z
Bρ (x0 )
¬2 ± 1
¬
2
¬
¬
¬σR − (σR )x0 ,ρ ¬ dx

h ° ρ ±2 

≤ c2
R 
Z
B R (x0 )
2
 12
¬2 
¬
i
¬
¬
2
³
²

¬σR − (σR )x0 , R ¬ dx + R kf k ∞
2
L
B R (x0 )
2
234
M. Fuchs, G. Seregin / A twodimensional variational model
being valid for any 0 < ρ ≤
À Z
Bρ (x0 )
R
,
2
hence
! 21
¬2
¬
¬
¬
¬σ − (σ)x0 ,ρ ¬ dx
≤ c3
"
° ρ ±2
R
À Z
! 12
2
|σ − (σ)x0 ,R | dx
+ R2 +
À Z
2
! 12 #
|σ − σR | dx
(6.8)
B R (x0 )
BR (x0 )
2
which is true for ρ ≤ R. We are going to discuss the second integral on the right-hand
side of (6.8):
1
2µ
Z
|σ − σR |2 dx
B R (x0 )
2
Z
1
=
2µ
(6.7)
B R (x0 )
=
Z
1
2µ
ε(uR ) : (σ − σR ) dx
B R (x0 )
2
(6.7)
Z
1
σ : (σ − σR ) dx −
µ
2
σ : (σ − σR ) dx
B R (x0 )
2
Z
=
ε(u) : (σ − σR ) dx +
B R (x0 )
(6.7)
=
³
1
σ − ε(u) : (σ − σR ) dx
2µ
B R (x0 )
2
Z
²
Z
2
²
³
1
σ − ε(u) : (σ − σR ) dx
2µ
B R (x0 )
2
²
Z
=
³
1
σ − ε(u) : (σ − σR ) dx.
2µ
)∪Ω31 (x0 , R
)
ω 12 (x0 , R
2
2
This implies
Z
B R (x0 )
2
2
|σ − σR | dx
≤
2
(2µ)
Z
)∪Ω31 (x0 , R
)
ω 12 (x0 , R
2
2
¬
¬2
¬1
¬
¬ σ − ε(u)¬ dx.
¬ 2µ
¬
(6.9)
M. Fuchs, G. Seregin / A twodimensional variational model
235
As in Section 5 we decompose
ω 12 (x0 , R2 )
¨ 12
ω− (x0 , R2 )
¨ 31
ω+ (x0 , R2 )
¨ 12
ω− (x0 , R2 )
¨ 12
ω0 (x0 , R2 )
¨ 31
ω0 (x0 , R2 )
¨ 31
ω0 (x0 , R2 )
∪
∩
∩
∪
∩
∩
∩
Ω31 (x0 , R2 )
©
23
(x0 , R2 )
ω+
©
23
(x0 , R2 )
ω−
¡
31
(x0 , R2 )
ω+
©
23
(x0 , R2 )
ω+
©
23
(x0 , R2 )
ω−
ω012 (x0 , R2 )
=
∪
∪
©
ω023 (x0 , R2 )
∩
∪
(6.10)
∪
∪
©
ω023 (x0 , R2 )
∩
¬
¬
¬1
¬
and observe that according to (2.8) the quantity ¬ 2µ
σ − ε(u)¬ is bounded on the last three
sets which occur on the right-hand side of (6.10). Using (2.8) also on the remaining sets
we find that
Z
)∪Ω31 (x0 , R
)
ω 12 (x0 , R
2
2
¬
¬2
¬1
¬
¬ σ − ε(u)¬ dx
¬ 2µ
¬
≤ c4
"
Z
ω 12 (x0 , R
)∪Ω31 (x0 , R
)
2
2
≤ c5
"
Z
ω 12 (x0 , R
)∪Ω31 (x0 , R
)
2
2
¬
¬#
¬
R
R ¬
|σ|2 dx + ¬¬ω 12 (x0 , ) ∪ Ω31 (x0 , )¬¬
2
2
¬ ¬
¬³ #
²
¬
¬
¬2
¬2 ³ ²¬¬
¬
¬
¬
R
R
¬
¬
¬
¬
¬ω 12 (x0 , )¬ + ¬Ω31 (x0 , )¬ .
¬σ − (σ)x0 , R2 ¬ dx+ 1 + ¬(σ)x0 , R2 ¬
¬
¬
¬
2
2 ¬
From assumption (6.1) it follows that
¬
¬
¬
¬
¬(σ)x0 , R2 ¬
≤
2R1 +
2|ε0 |
.
a
Now, by combining the last two estimates and using Hölder’s inequality, we get
Z
)∪Ω31 (x0 , R
)
ω 12 (x0 , R
2
2
¬
¬2
¬1
¬
¬ σ − ε(u)¬ dx
¬ 2µ
¬
"
|ω 12 (x0 , R2 )| |Ω31 (x0 , R2 )|
+
|B R (x0 )|
|B R (x0 )|
2
2
 p2

À
!1− p2
#
Z
¬p 
¬

|ω 12 (x0 , R2 )| |Ω31 (x0 , R2 )|
 − ¬¬σ − (σ) R ¬¬ dx . (6.11)
+
+
x0 , 2


|B R (x0 )|
|B R (x0 )|
≤ c6 R 2
2
2
B R (x0 )
2
236
M. Fuchs, G. Seregin / A twodimensional variational model
The measures |ω 12 (x0 , R2 )| and |Ω31 (x0 , R2 )| have already been estimated in Section 5 with
the result
¬ ¬
¬
¬−1
¬
¬ 12
¬ω (x0 , R )¬ ≤ ¬¬∆12 R ¬¬
¬
x0 , 2
2 ¬
Z
¬
¬ ¬
¬−1
¬ 31
¬
¬Ω (x0 , R )¬ ≤ ¬¬∆31 R ¬¬
¬
x0 , 2
2 ¬
Z
ω 12 (x0 , R
)
2
¬
¬
¬
¬
¬σ − (σ)x0 , R2 ¬ dx,
)
Ω31 (x0 , R
2
¬
¬
¬
¬
¬σ − (σ)x0 , R2 ¬ dx.
From Hölder’s inequality we infer
À
|ω 12 (x0 , R2 )|
|B R (x0 )|
! p1
2
 p1
Z
¬
¬p 
¬−1 
¬
¬
¬
¬ 
¬

R
−
≤ ¬∆12
σ
−
(σ)
¬ 
¬
x0 , 2 ¬ dx ,
x0 , R
2

B R (x0 )
2
À
|Ω31 (x0 , R2 )|
|B R (x0 )|
! p1
2
 p1

(6.12)
Z ¬
¬
¬p 
¬
¬ 31 ¬−1 
¬
¬

R
−
≤ ¬∆x0 , R ¬ 
σ
−
(σ)
¬
x0 , 2 ¬ dx .

2
B R (x0 )
2
In a final step we apply Poincare’s inequality and use (5.3) to get
 p2

Z ¬
Z
¬p 

 − ¬¬σ − (σ) R ¬¬ dx ≤ c7
x0 , 2


B R (x0 )
2
|∇σ|2 dx
(6.13)
B R (x0 )
2
£
¡ 2
¢ −2
−2
Ψ (xo , R) + R2 .
+ |∆31
≤ c8 1 + |∆12
x0 ,R |
x0 ,R |
The desired claim (6.2) is then a consequence of (6.8), (6.9) and (6.11)–(6.13). Since the
proofs of (6.4) and (6.6) are less difficult than the proof of (6.2), we just present the ideas
for (6.4), the remaining claim (6.6) is left to the reader. Again we consider the auxiliary
M. Fuchs, G. Seregin / A twodimensional variational model
237
problem (6.7) and use estimate (6.8). Then we proceed as follows: we have
Z
1
|σ − σR |2 dx
2
B R (x0 )
2
Z
1
=
2
σ : (σ − σR ) dx −
B R (x0 )
1
=
2
ε(uR ) : (σ − σR ) dx
B R (x0 )
2
Z
Z
2
σ : (σ − σR ) dx
B R (x0 )
2
=
Z
(ε(u) − ε0 ) : (σ − σR ) dx +
B R (x0 )
³
1
σ − (ε(u) − ε0 ) : (σ − σR ) dx
2
2
²
Z
²
B R (x0 )
2
=
Z
³
1
σ − (ε(u) − ε0 ) : (σ − σR ) dx.
2
B R (x0 )
2
Applying Hölder’s inequality and using (2.8)
Z
Z
2
|σ − σR | dx ≤ 4
B R (x0 )
we find that
¬2
¬
¬
¬1
¬ σ − (ε(u) − ε0 )¬ dx
¬
¬2
B R (x0 )
2
2
¬
¬2
¬1
¬
¬ σ − (ε(u) − ε0 )¬ dx.
¬2
¬
Z
=4
ω 23 (x0 , R
)∪Ω12 (x0 , R
)
2
2
¬
¬
From (2.8) we deduce boundedness of ¬ 12 σ − (ε(u) − ε0 )¬ on ω 23 (x0 , R2 ) ∪ Ω12 (x0 , R2 ),
hence
¬ ¬
¬³
²¬
Z
¬
¬
¬
¬
R
R
(6.14)
|σ − σR |2 dx ≤ c9 ¬¬ω 23 (x0 , )¬¬ + ¬¬Ω12 (x0 , )¬¬ .
2
2
B R (x0 )
2
Analogous to estimate (6.12) we get
À
|ω 23 (x0 , R
)|
2
|B R (x0 )|
! p1
0 2
2
À
|Ω12 (x0 , R
)|
2
|B R (x0 )|
2
¬−1
¬
¬
¬
≤ ¬∆23
¬
x ,R
! p1
 p1
Z ¬
¬p 

 − ¬¬σ − (σ) R ¬¬ dx ,
x0 , 2



B R (x0 )
2

 p1
Z ¬
¬
¬
¬p 
¬ 12 ¬−1 
¬
¬


≤ ¬∆x , R ¬
 − ¬σ − (σ)x0 , R2 ¬ dx .
0 2
B R (x0 )
2
Using Poincare’s inequality together with the estimates (6.8), (6.14), (6.15) and recalling the Caccioppoli-type inequality (5.4), the proof of (6.4) and hence of Lemma 6.1 is
complete.
238
7.
M. Fuchs, G. Seregin / A twodimensional variational model
Proof of Theorem 2.4
In this section we are going to prove Theorem 2.4 for a representative case. So let us
assume that µ > 1 and g0 < 0 which means that formula (2.8) of Lemma 2.1 is valid. All
other cases can be treated in the same way.
Lemma 7.1. Consider numbers p > 2, ν ∈ (0, 1), γ > 0 and let Ω0 denote a subdomain
with compact closure in Ω.
Consider t ∈ (0, 1) such that
2 c1 t1−ν ≤ 1
(7.1)
where the constant c1 is defined in Lemma 6.1. Suppose that for some point x0 ∈ Ω0 and
some radius R < 12 R0 = 12 dist(Ω0 , ∂Ω) the following statements are true:
²
³
2p+2
t−(1+ν)
Ψ(x0 , R) + c1 1 + p R
γ
1 − t1−ν
≤ C(ν, t, p, γ) = min
À

2
t
 t + 2p+2
γp
2
! p−2


γ
ν
, , t γ(1 − t )
(7.2)

4
and either
31
∆12
x0 ,R ≥ 2 γ , ∆x0 ,R ≤ −2 γ ,
(7.31 )
12
∆23
x0 ,R ≥ 2 γ , ∆x0 ,R ≤ −2 γ ,
(7.32 )
23
∆31
x0 ,R ≥ 2 γ , ∆x0 ,R ≤ −2 γ .
(7.33 )
or
or
Then, for any k ∈ N0 , we get
Ψ(xo , tk R)
Ψ(xo , tk+1 R)
≤
tν(k+1)
≤
min
À

2
t
 t + 2p+2
γp
2
! p−2
"
,

γ
4
,
#
²
³ X
k
p+2
2
Ψ(x0 , R) + c1 t−(1+ν) 1 + p R
ts(1−ν)
γ
s=0
(7.4)
(7.5)
and either
(
12
∆12
xo ,R + γ ≥ ∆x0 ,tk R ≥ γ
31
∆31
xo ,R − γ ≤ ∆x0 ,tk R ≤ −γ,
(7.61 )
(
23
∆23
xo ,R + γ ≥ ∆x0 ,tk R ≥ γ
12
∆12
xo ,R − γ ≤ ∆x0 ,tk R ≤ −γ,
(7.62 )
(
31
∆31
xo ,R + γ ≥ ∆x0 ,tk R ≥ γ
23
∆23
xo ,R − γ ≤ ∆x0 ,tk R ≤ −γ,
(7.63 )
or
or
respectively.
M. Fuchs, G. Seregin / A twodimensional variational model
239
Proof. Lemma 7.1 follows by induction from Lemma 6.1 taking into account the inequalities
¬
¬
¬
¬ ij
ij
¬∆x0 ,ts+1 R − ∆x0 ,ts R ¬ ≤ |(σ)x0 ,ts+1 R − (σ)x0 ,ts R |
Z
≤
−
|σ − (σ)x0 ,ts R | dx ≤ t−1 Ψ(x0 , ts R).
Bts+1 R (x0 )
2
For a detailed computation in a similar setting we refer to [20, 21].
In order to prove Theorem 2.4 we first observe that
©
31
<
0
∪
>
0,
∆
a(σ) = {x ∈ Ω0 : ∆12
x
x
©
12
{x ∈ Ω0 : ∆23
x > 0, ∆x < 0 ∪
©
23
{x ∈ Ω0 : ∆31
x > 0, ∆x < 0 ,
(7.7)
Ω0 denoting the Lebesgue set of σ. Next take some x0 ∈ a(σ), w.l.o.g. we may assume
that
©
¨
31
x0 ∈ x ∈ Ω0 : ∆12
x > 0, ∆x < 0 ,
hence we have
lim ∆12
x0 ,R = γ12 > 0,
R↓0
lim ∆31
x0 ,R = γ31 < 0.
R↓0
(7.8)
Next we consider a subdomain Ω0 ⊂⊂ Ω such that x0 ∈ Ω0 and define the numbers p > 2
and γ > 0 according to
2γ < min{γ12 , −γ31 }.
(7.9)
For any ν ∈ (0, 1) we select a number t with (7.1). By Poincare’s inequality (compare
(6.13)) we know
Ψ(x, R) −→ 0
(7.10)
R↓0
for any x in Ω. According to (7.8)–(7.10) there exists a radius 0 < R <
1
dist(∂Ω, Ω0 ) such that
2
²
³
2p+2
t−(1+ν)
Ψ(x0 , R) + c1 1 + p R
< C(ν, t, p, γ),
γ
1 − t1−ν
∆12
x0 ,R
> 2γ,
∆31
x0 ,R
1
R
2 0
=
(7.11)
< −2γ.
31
Since the functions x 7→ Ψ(x, R), x 7→ ∆12
x,R , x 7→ ∆x,R are continuous, we find a neighborhood U of x0 such that (7.11) holds for any x ∈ U . Then we deduce from Lemma 7.1
that σ is Hölder continuous near x0 with exponent ν, moreover, (7.61 ) implies that this
31
neighborhood belongs to the set {x ∈ Ω0 : ∆12
x > 0, ∆x < 0}. This completes the proof
of Theorem 2.4.
240
M. Fuchs, G. Seregin / A twodimensional variational model
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