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Page 1: MPI MIS Leipzig, Preprint Nr. 3

Max�Planck�Institut

f�ur Mathematik

in den Naturwissenschaften

Leipzig

Microstructures� phase transitions

and geometry

by

Stefan M�uller

Preprint�Nr�� � ����

Page 2: MPI MIS Leipzig, Preprint Nr. 3
Page 3: MPI MIS Leipzig, Preprint Nr. 3

Microstructures� phase transitions and geometry

Stefan M�ullerMax�Planck�Institute for Mathematics in the Sciences

Inselstr� ������ ����� Leipzig� Germany

Contents

� What are microstructures� �

� Microstructure in elastic crystals �

� Basic mathematical questions �

� The two�gradient problem �

��� Exact solutions � � � � � � � � � � � � � � � � � � � � � � � � � � ���� Approximate solutions � � � � � � � � � � � � � � � � � � � � � � ���� Relaxation� nonattainment� microstructure � � � � � � � � � � � �

� Approximate solutions and Young measures �

� Exact solutions and convex integration ��

��� Existence of solutions � � � � � � � � � � � � � � � � � � � � � � � ����� Regularity and rigidity � � � � � � � � � � � � � � � � � � � � � � �

� Length scales surface energy and singular perturbations �

�� A one dimensional model � � � � � � � � � � � � � � � � � � � � � ���� Surface energy and domain branching � � � � � � � � � � � � � � ��

� Outlook ��

�� Beyond Young measures � � � � � � � � � � � � � � � � � � � � � ���� Dynamics � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� Computation � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 4: MPI MIS Leipzig, Preprint Nr. 3

� What are microstructures�

Microstructures are structures on a scale between the atomic scale and themacroscopic scale on which we usually make observations� They are abun�dant in natural and man�made materials� and often the microstructure op�timizes a material s properties �maximum strength at given weight� minimalenergy� maximum or minimum permeability� ����� Some materials can changetheir internal microstructure and hence their properties in response to exter�nal in�uences� They are sometimes referred to as �smart materials and areof great technological interest�

The attempt to mathematically describe and analyze the formation� in�teraction and the macroscopic e�ects of microstructures yields new� easilystated but deep mathematical problems� Their resolution is in its very be�ginning and involves the interaction of a variety of branches of mathematicsincluding the calculus of variations� di�erential geometry� geometric measuretheory� dynamical systems and nonlinear partial di�erential equations� Inthe following I try to describe some of the basic questions and how di�erentbranches of mathematics are involved in their resolution�

To simplify the exposition I focus on models of microstructures that arisein elastic crystals due to a solid�solid phase transformation� A large partof this article� in particular Sections ���� mainly reviews work of others� Idraw heavily on the beautiful survey of �Sver�ak �Sv ��� as well as the detailedexpositions of Ball and James �BJ �� �BJ ���� Currently no exposition ofthe �eld in book form is available but various projects are in preparation�BJ �a�� �BJ �b�� �Mu �a���Mu �b�� �PZ ��� Dacorogna s book �Da �� isa good reference for the direct method and notions of convexity�

� Microstructure in elastic crystals

Inspired by ideas of Ericksen� Ball and James �BJ � �see also �CK ���Fo ��� proposed to study crystal microstructure by analyzing minimizersand minimizing sequences for the elastic energy� They use a continuum modelbased on nonlinear elasticity �a theory based on linearized elasticity was pro�posed earlier by Khachaturyan� Roitburd and Shatalov� see �Kh ��� �Kh ����KSh ���� �Ro ���� �Ro �� a comparison of the two approaches appears in�BJ ���� Section �� �Bh ��� and �Ko ���� The elastic energy I needed to de�form the crystal from its reference con�guration �identi�ed with a bounded

Page 5: MPI MIS Leipzig, Preprint Nr. 3

domain � � R�� by a map u � �� R� is given by

I�u� �R

W �Du�dx� �����

where Du � � �ui

�xj� is the deformation gradient� The Cauchy�Born rule asserts

that the stored�energy density W �F � is the energy �per unit reference vol�ume� needed to perform an a�ne deformation x �� Fx of the crystal lattice�The stored energy is invariant under rigid transformations of the ambientspace �frame indi�erence� and under changes of the independent variablesthat correspond lattice invariant rotations �crystal symmetry�

W �QF � � W �F � �Q � SO���� �����W �FR� � W �F � �R � P � SO���� �����

The discrete point group P re�ects the symmetry properties of the latticethat persist in the continuum approach �see �Er ��� �Er ��� �Pi �� for therelevance of P and �Er ��� �Za ��� for a discussion of the Cauchy�Born rule��

It is convenient to normalize W such that minW � �� Then the set

K � fF � W �F � � �g

containts exactly the zero�energy a�ne deformations of the lattice� By �����and ����� this set consists of one or several disjoint copies of SO���

K � SO���U� � � � � � SO���Um� Ui � UTi � �� �����

In general the energy W and hence the set K depend on the temperature�� For materials that undergo solid�solid phase transitions K consists of onecomponent at high temperatures and of several symmetry related �throughP� components below a critical temperature �c �see �BJ �����The relation between minimization of I and microstructure is discussed inSection � for a model problem� situations with SO�n� symmetry are discussedin Section ��

� Basic mathematical questions

The fundamental problem is�

Characterize minimizers and minimizing

Page 6: MPI MIS Leipzig, Preprint Nr. 3

sequences of I�u� �RW �Du�dx

subject to suitable boundary conditions�

This is a very di�cult problem since W is typically not convex �nor rank��convex� for materials that undergo solid�solid phase transitions� In otherwords the corresponding Euler�Lagrange equations are not strongly ellipticbut of mixed elliptic�hyperbolic type� Research has therefore mostly focusedon �almost� zero energy states� i�e� pointwise minimizers of the integrand�Generalizing the setting slightly we consider maps

u � � � Rn � Rm

on a bounded domain � �with Lipschitz boundary� and a compact set K �Mm�n in the m� n matrices �e�g� the zero set of W ��

Problem � �exact solutions�� Characterize all Lipschitz maps u such that

Du � K a�e� in �� �����

Problem � �approximate solutions�� Characterize all sequences uj ofmaps whose Lipschitz constants are uniformly bounded and which satisfy

dist�Duj� K�� � a�e� in �� �����

Problem � �relaxation ofK�� Determine the sets Kex andKapp � Mm�n

of all matrices F such that Problem � and Problem � have a solution thatsatis�es in addition

u�x� � Fx on ��� �����uj�x� � Fx on ��� �����

respectively�A simple covering and scaling argument shows that Kex and Kapp do not

depend on ��In the context of crystal microstructures the sets Kex and Kapp have an

important interpretation� They represent macroscopic zero �or almost zero�energy deformations� while K represents microscopic zero energy deforma�tions� The sets Kex and Kapp can be much larger than K� in fact it isconjectured �and proved in two dimensions� that for many sets of the form

Page 7: MPI MIS Leipzig, Preprint Nr. 3

����� the sets Kex and Kapp have interior points �subject to an incompress�ibility constraint detF � ��� This would correspond to an �experimentallyobserved� �uid�like behavior� Some known results are reviewed in Section �below�

Problems ��� also arise in many other contexts� e�g� in the classi�cationof isometric immersions or in the theory of nonlinear elliptic and hyperbolicpartial di�erential equations� see �Sver�ak s survey �Sv ��� and Gromov s trea�tise �Gr ���

If minimizers do not exist there are often many di�erent minimizing se�quences� Some of those� however� only di�er in rather super�cial ways�

Problem �� Develop mathematical objects that capture the �essentialfeatures of minimizing sequences�

One such object is the Young measure� discussed in Section �� see alsoSection ��� Minimization of elastic energy often predicts minimizing se�quences that develop �ner and �ner oscillations� Experimentally a limited�neness is observed�

Problem �� Explain the length scale and �ne geometry of microstruc�tures� possibly by including other contributions to the energy� such as inter�facial energy�

Some attempts in this directions are discussed in Section �

� The two�gradient problem

Before reviewing a general framework for Problems ��� let us look at thesimplest example

K � fA�Bg�

��� Exact solutions

The simplest solution satis�es Du � A on a halfspace H and Du � Bon its complement� Due to tangential continuity of u this is only possibleif B � A � a � n� where n is the normal to H� In particular one hasrk�B � A� � �� It turns out that for an arbitrary solution of Du � K� thegradient can only jump across hyperplanes with normal n�

Page 8: MPI MIS Leipzig, Preprint Nr. 3

Lemma ��� ��BJ ����� Suppose that u � Rn � Rm is Lipschitz and

Du � fA�Bg a�e�

�i� If rk�B � A� � � then Du A a�e or Du B a�e

�ii� If rk�B � A� � � then B � A � a � n and there exists a Lipschitzfunction h � R� R such that h� � f�� �g a�e and

u�x� � Ax� a h�x n��

The proof is a simple exercise using the fact that the distributional curl of agradient vanishes�

��� Approximate solutions

Lemma ��� ��BJ ����� Suppose that rk�B � A� � � and let uj � � � Rn

be a sequence of functions with uniformly bounded Lipschitz constants thatsatisfy

dist�Duj� fA�Bg�� � a�e� in ��

Then �for a subsequence� not relabelled� either

Duj � A a�e� or Duj � B a�e�

This can be proved e�g� by using the minors relations �see Section ���

��� Relaxation nonattainment microstructure

If rk�B � A� � � then Kapp � K by Lemma ����

Lemma ��� ��BJ ����� Suppose that rk�B � A� � �� Then

�i� Kex � fA�Bg�

�ii� Kapp � convK � f�A� ��� ��B � � � ��� ��g�

Moreover for any sequence uj� that satis�es ��� and ���� one has

uj� � Fx uniformly on �� �����

Page 9: MPI MIS Leipzig, Preprint Nr. 3

Dv=Bλ

Dv=A

A

B

A

n

B - A = a x n

1−λ

Figure �� A simple laminate�

In particular for � � ��� ��� F � �A� ��� ��B there exist approximate butnot exact solutions� The gradients Duj of minimizing sequences must oscil�late rapidly so that uj can approach Fx� Nonexistence of a minimizer thusleads to �ne scale oscillations and �microstructure �

We sketch one possible construction of a minimizing sequence� Many vari�ants are possible� and we return in Section � to the question whether allminimizing sequences are unique up to minor details�

Assume without loss of generality that F � �� let h be a ��periodic Lipschitzfunction with h� � ���� �� on ��� ��� h� � � on ��� �� and let

v�x� � h�x n�� vk�x� � k��v�kx��

The functions vk are called simple laminates since the sets fDu � Ag andfDu � Bg consist of parallel stripes �see Figure ��� We have Dvk � fA�Bga�e and vk � � uniformly� To adjust the boundary conditions consider thecut�o� functions �k�x� � minfkdist�x� ���� �g� Then uk � �kvk has thedesired properties�

Page 10: MPI MIS Leipzig, Preprint Nr. 3

A B A B

boundary layer of thickness 1/k

(1−λ) / k

λ / k

Figure �� A minimizing sequence obtained by truncation of very �ne lami�nates�

� Approximate solutions and Young measures

Young measures �also called �generalized functions � were introduced byL�C� Young in the �� s to solve optimal control problems that have no classi�cal solution but rather require an in�nitesimally fast switching between states�see �Yo ��� �Yo ����� The idea is to replace functions which take values ina �compact� set K by functions that take values in the space of probabilitymeasures on K� Thus an �in�nitesimally �ne mixture of two states is rep�resented by a convex combination of two Dirac masses� For the problemsdiscussed above it is crucial to characterize those Young measures that aregenerated by sequences of gradients� An abstract characterization is avail�able in terms of quasiconvex functions� but already simple concrete exampleslead to challenging open questions� since very few quasiconvex functions areknown�The presentation below follows �Ba ��� A short selection of the rich litera�

ture on Young measures is �Ba ��� �BL ��� �Kr ���� �Va ���� �Va ���� Tartar�Ta ��� �Ta �� and later DiPerna �DP �� used Young measures in connec�tion with the theory of compensated compactness to analyze compactnessand existence questions in nonlinear partial di�erential equations� Varifolds

Page 11: MPI MIS Leipzig, Preprint Nr. 3

��Al ��� are a generalizaton of Young measures to a geometric setting�

Let E be a measurable set in Rn� K a compact set in Rl and denoteby M�K� and P�K� the space of the Radon measures and of probabilitymeasures on K� The pairing between the space C�K� of continuous func�tions and its dual M�K� is denoted by h�� fi �

RKfd�� A Young measure

� � E � P�K� is a weak�measurable map� i�e� a map such that x �� h��x�� fiis measurable for all f � C�K��

Theorem ��� �Fundamental theorem on Young measures�Let wj � E � K be a sequence of measurable functions� Then there existsa subsequence wjk and a Young measure � � E � P�K� such that for allf � C�K�

f�wjk��� f� in L��E�� f��x� � h�x� fi �

RKfd�x� �����

Moreover if K � � K is a compact subset then�

supp�x � K � for a�e� x� dist �wjk� K�� � �locally� in measure� �����

Notation� We say that the sequence wjk generates the Young measure �� AYoung measure � is called a �W �����gradient Young measure if E is openand there exists a sequence uj Lipschitz functions with uniformly boundedLipschitz constants such that Duj generates ��

Remark� A similar result holds for sequences wj � E � Rl if onereplaces C�K� by C��R

l�� the dual of M�Rl�� In this case� however� ��x� isin general only a subprobability measure� Mild additional condition such assupj

RE

jwjjs � for some s � � assure that ��x� is a probability measure�

Alternatively one can work in a suitable compacti�cation of Rl�

Proof� To each wj we associate the elementary Young measure

�j�x� � wjx��

The sequence f�jgj�N is bounded in the space L�w �E�M�K�� of essentiallybounded� weak� measurable maps� Since this space is a dual of L��E�C�K���see �Ed ���� p� �� �IT ���� p� ��� �Me ���� p� ���� the �rst assertion followsfrom the Banach�Alaoglu theorem and the fact that P�K� is weak� closed inM�K�� The second assertion is a simple consequence of the �rst�

Page 12: MPI MIS Leipzig, Preprint Nr. 3

Young measures are closely related to compactness by the following fact�

Corollary ��� Let wjk be as in Theorem ��� Then

wjk � w locally in measure � ��x� � wx� a�e�

Young measures provide a concise description of the �ne scale oscillation thatmay arise in minimizing sequences�

Example ��� As in Section ��� consider A�B� F � Mm�n with rk�B �A� � �� F � �A � �� � ��B� � � ��� ��� Let uj be a sequence withdist�Duj� fA�Bg� � � a�e� uj�x� � Fx on ��� Then Duj generates the�unique� Young measure �A � ��� ��B�

Proof� By ������ supp ��x� � fA�Bg� Hence ��x� � ��x�A������x��B�Application of ����� with f � id and Lemma ��� �ii� yield ��x�A � �� ���x��B � F � which implies the assertion�

Young measures can be seen as generalized solutions of the problem Du �

K orRW �Du�

�� min generated by minimizing sequences� A classical so�

lution corresponds to a Young measure that is a Dirac mass at every point�Problem � �approximate solutions� can be rephrased as follows�

Problem ��� Characterize all gradient Young measures supported in K�

To classify gradient Young measures Morrey s notion of quasiconvexity isfundamental�

De nition ��� A function f � Mm�n � R is quasiconvex if for every

bounded domain � and every F �Mm�n

Rf�F �D�� �

Rf�F � � j�jf�F �� �� Lipschitz � �j� � �� �����

Thus f is quasiconvex if a�ne functions minimizeRf�Du� subject to a�ne

boundary values� The de�nition is independent of �� and quasiconvex func�tions are automatically continuous�Morrey s crucial observation was that quasiconvexity of f is equivalent

to �sequential� W ��� weak� lower semicontinuity of the functional u ��Rf�Du�dx� Combining this fact with careful measure�theoretic and func�

tional�analytic arguments� Kinderlehrer and Pedregal obtained the followingclassi�cation�

��

Page 13: MPI MIS Leipzig, Preprint Nr. 3

Theorem ��� ��KP ���� A Young measure � � �� P�Mm�n� is a �W �����gradient Young measure if and only if the following three conditions hold

�i� supp �x is uniformly bounded �for a�e� x��

�ii� h�x� idi is the gradient of a Lipschitz function�

�iii� f�h�x� idi� � h�x� fi for all quasiconvex f �

The key point is �iii� which is in nice duality with De�nition ���� Roughlyspeaking� quasiconvex function satisfy Jensen s inequality for gradients whilegradient Young measures must satisfy Jensen s inequality for all quasiconvexfunctions� A similar result holds for Young measures generated by gradientsof sequences that are bounded in the Sobolev spaceW ��p�p � ��� see �KP �����Kr ���� �FMP ��� general references on Young measures for noncompacttargets include �DPM � and �Ro ����As a consequence we obtain an abstract solution of Problem �� and Prob�

lem �� A Young measre � is called homogeneous if the map x �� ��x� isconstant �a�e��� A blow�up argument shows that if � is a gradient Youngmeasure then ��a� arises as a homogeneous gradient Young mesure for a�e�a � �� see �KP ����

Corollary ��� Let K �Mm�n be compact�

�i� The set of homogeneous gradient Young measures supported on K isgiven by

Mqc�K� �� f� � P�K� � f�h�� idi� � h�� fi �f �Mm�n � R quasiconvexg�

�ii� The set Kapp is the quasiconvex hull of K�

Kapp � Kqc �� fF � f�F � � infK f� �f � Mm�n � R quasiconvexg� fh�� idi � � � Mqc�K�g �

Following �Sver�ak we say that Mqc�K� is trivial if it only contains Diracmasses� In view of Corollary ��� this occurs if and only if all approximatingsequences are compact�Quasiconvexity is a natural condition� but hard to verify� Therefore two al�gebraic conditions were introduced� A function f �Mm�n � R is polyconvex

��

Page 14: MPI MIS Leipzig, Preprint Nr. 3

if it can be expressed as a convex function of minors �subdeterminants�� itis called rank���convex if it is convex on all line segments �A�B� with rankB � A � �� One has the implications ��Mo ���� �Da ����

f polyconvex f quasiconvex f rank�� convex�

For m � � or n � � all notions are equivalent to ordinary convexity� Form�n � � quasiconvexity does not imply polyconvexity�

Conjecture ��� �Morrey� �� �� For m�n � � rank�� convexity does notimply quasiconvexity�

�Sver�ak �Sv ��a� proved the conjecture for m � �� The case m � � is open�Kristensen �Kr �� very recently proved that for m � � there is no localcondition that implies quasiconvexity�For a minorM application of Theorem ��� to�M yields the minors relation

h�x�Mi � M�h�x� idi� �����

as a necessary condition for gradient Young measures� The following ta�ble illustrates some solved and unsolved problems for homogeneous gradientYoung measures�

��

Page 15: MPI MIS Leipzig, Preprint Nr. 3

K

Nontrivial

Nontrivial

Method

References

exactsoln�

GYM

fA�Bg

No

No

Minorsor

�BJ���

Cauchy�Riemanneqns�

�Sv��a�

fA�B�Cg

No

No

Nonconvexsolns�

�Sv��b�

ofMonge�Amp�ere

�Sv�b�

nite

Yes

Examplewith

�AH��� �Ta���

fourmatrices

�CT��� �BFJK���

countable

Yes

Yes

Example

�Sv�

SO���RSO��

No�

No

Cauchy�Riemann

�Sv��a�

holomorphicfns�

eqns�

SO�n��RSO�n�

No�

No

Minors�degenerate

�Ki���

n��

M�obiusmaps

No

ellipticeqns�

�Re���

�ki

�� SO��Ai

No

No

Minorsor

�Sv���

ellipticregularity

SO���A�SO���B

Specialcasesknown

�Ma��� �Sv���

scalarconservation

No

No

compensated

�Ta���

laws

compactness�

div�curllemma

m�elliptic

Yes m

��

�Sv���

systems

m

��

Table��NontrivialexactsolutionsandnontrivialgradientYoungmeasures�GYM�ofincompatible

setsK

�i�e�rk�A�B�����A�B�K��see�Sv��� �BFJK���andthereferencesthereinforfurther

examples�

��

Page 16: MPI MIS Leipzig, Preprint Nr. 3

� Exact solutions and convex integration

Approximate solutions are characterized by the quasiconvex hull Kqc andset Mqc�K� of Young measures� The construction of exact solutions is moredelicate� In view of the negative result for the two�gradient problem �seeLemma ��� �i�� it was widely believed that exact solutions are rather rare�Recent results suggest that many exact solutions exist but that they haveto be very complicated� This is reminiscent of rigidity and �exibility resultson isometric immersions and other geometric problems �see �Na ���� �Ku �����Gr ��� Section ��������

To illustrate some of the di�culties consider the two�dimensional two�wellproblem�

Du � K a�e� in �� u � Fx on ��� �����

K � SO���A � SO���B� �����

A � Id � B � diag��� ��� � � � �� �����

If we ignore boundary conditions the simplest solutions of Du � K aresimple laminates� see Figure �� A short analysis of the rank�� connections inK shows that such laminates are perpendicular to one of the normals n� orn�� determined by the two solutions of the equation

QA� B � a� n� �����

There is� however� no obvious way to combine the two laminates �see Fig�ure ��� It was thus believed that the problem ����� ����� has no nontrivialsolutions� This is false� The construction of nontrivial solutions is based onGromov s method of convex integration�

��� Existence of solutions

First� one observes that the open version of the two�gradient problem admitsa solution�

Lemma ��� ��MS ����� Suppose that rk�B�A� � �� F � �A������B� � ���� ��� Then� for a bounded domain � and every � � � there exists a piecewise

��

Page 17: MPI MIS Leipzig, Preprint Nr. 3

n 1

Q B 1

A

Q B 1

An 2

A

Q B 2

A

Q B 2

Figure �� Two possible laminates for the two�well problem�

A

Q B 1

Q B 2

Q B 1

Q A 2

Q B 3

A

Figure �� None of the above constructions satis�es the rank�� condition acrossevery interface�

��

Page 18: MPI MIS Leipzig, Preprint Nr. 3

linear map u such thatu�x� � Fx on ��

dist�Du� fA�Bg� ��

sup ju�x�� Fxj ��

Here a map u � �� Rm is called piecewise linear if u is Lipschitz continuousand if there exist �nite or countably many disjoint open sets �i whose unionhas full measure such that uji

is a�ne� To iterate the result of Lemma ���we de�ne the lamination convex hull K l of a set K � Mm�n by successiveaddition of rank�� segments�

K� �� K�Ki �� �� Ki� � f�A� ��� ��B� � � ��� ��� A� B � Ki�� rk�B � A� � �g

K l ���Si��

Ki��

Remark� In contrast with the ordinary convex hull there is no versionof Carath�eodory s theorem that guarantees that the union of a �xed numberof the Ki� is su�cient� Generally� very little is known about the geometryof rank�� convexity �but see �MP �����

Lemma ��� Suppose that U �Mm�n is open� Let v � �� Rm be piecewisea�ne and Lipschitz continuous and suppose Dv � U l a�e� Then there existu � �� Rm such that

Du � U a�e� in �� u � v on ���

The crucial step is the passage from open to compact sets K � Mm�n�Following Gromov we say that a sequence of sets Ui is an in�approximationof K if

�i� the Ui are open and contained in a �xed ball

�ii� Ui � U li �

�iii� Ui � K in the following sense� if Fik � Uik � ik �� and Fik � F � thenF � K�

��

Page 19: MPI MIS Leipzig, Preprint Nr. 3

Theorem ��� ��Gr ���� p� ��� �MS ����� Suppose that K admits an in�approximation fUig� Let v � C����Rm� with

Dv � U��

Then there exists a Lipschitz map u such that

Du � K � � a�e�� u � v on ���

The proof uses a sequence of approximations obtained by successive applica�tion of Lemma ���� To achieve strong convergence each approximation usesa much �ner spatial scale than the previous one� similar to the constructionof continuous but nowhere di�erentiable functions� This is one of the keyideas of convex integration�

This theorem applies to the two�well problem ����� ����� if �� �� �� Acalculation shows that solutions exist if F � int K l� and K l is given by asimple formula �see �Sv ����� A di�erent construction of �many� nontrivialsolutions was recently obtained by Dacorogna and Marcellini �DM ��b� usingBaire s theorem� Other existence results appear in �DM ��a�� �DM ��� Thecase �� � � in ����� ����� requires �i� of the following re�nement�

Theorem ��� ��MS ����� Theorem �� remains valid if the de�nition ofin�approximation is modi�ed as follows�

�i� the sets Ui are relatively open in the set fF �Mn�n � detF � �g� n � ��

�ii� the lamination convex hull U l is replaced by the rank�� convex hull

U rc �� fF �Mm�n � f�F � � infUf for

all rank�� convex f � Mm�n � Rg�

Part �i� just requires only a modi�cation in the proof of Lemma ���� A com�pletely open problem is how many constraints on F can be prescribed� Anatural guess is that all minors can be prescribed but this is false�

Regarding part �ii� one trivially has the inclusion K l � Krc� Several au�thors ��AH ��� �CT ���� �Ta ���� discovered independently an example whereKrl �� Krc� Let

K �

��

�� �

� �

���

�� �

� �

��

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one has K l � K but

Krc �

��� �

� �

�� j�j � �� j�j � �

��

It follows from Theorem ��� �ii� that the relation Du � U admits nontrivialsolutions for every �small� neighbourhood U of K� although U contains norank�� connections �i�e� no simple laminates solve the relation��A natural question is whether one can use the quasiconvex hull U qc insteadof U rc� One key point seems to be the resolution of the following�

Conjecture ��� Let K be a compact quasiconvex set� i�e� Kqc � K andlet � � Mqc�K�� Then for every open set U � K there exists a sequenceuj � ��� ��

n � Rm such that Duj generates � and Duj � U a�e�

The conjecture is true for compact convex sets ��Mu �a��� this re�nes a re�sult of Zhang �Zh ��� who showed that Duj � B��� R� for a su�ciently largeball�

The use of the rank�� convex hull U rc rather then U l is potentially relevantfor the construction of singular solutions to m� � elliptic systems

div��Dv� � �� v � � � R� � Rm�

Such a system may be written as �rst order system �see �Sv ����

Du � K �� f�F�G� �M�m��� ��F � � G�g�

where G�i� � �Gi�� G

�i� � Gi�� Ellipticity implies thatK l � K� but nontrivial

solutions may exist if one can show Krc �� K�

��� Regularity and rigidity

The construction outlined above yields very complicated solutions of the two�well problem ����� ������ This raises the question whether the geometry ofthe solutions can be controlled� Consider the set

E � fx � � � Du�x� � SO���Ag

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n 2

n 1

Figure �� Structure of solutions with �nite perimeter� The normals n�� n�are determined by ������

where Du takes values in one connected component of K �or one phase inthe applications to crystals�� The perimeter of a set E � � � Rn is de�nedas

PerE � supf

ZE

div � dx � � � C�����R

n�� j�j � �g�

For smooth or polyhedral sets this agrees with the �n� �� dimensional mea�sure of �E�

Theorem ��� ��DM � ��� If u is a solution of ����� � ���� and if PerE �then u is locally a simple laminate and �E consists of straight line segmentsthat can only intersect at ���

The proof combines geometric and measure�theoretic ideas� The geometricidea is that the Gauss curvature K�g� of the pull�back metric g � �Du�TDushould vanish �in a suitable sense�� Since g only takes two values this shouldgive information on E�To implement this idea one �rst establishes a version of Liouville s theorem�in�nitesimal rotations on an open connected set are rigid motions� for sets of�nite perimeter� The analogue of connected components in this frame workare indecomposable components �a set F of �nite perimeter is indecompos�able if for every F� � F with PerF � PerF� � Per�F n F�� the set F� orF n F� has zero measure�� To �nish the proof one decomposes Du as ei�g���

�with the usual identi�cation C � R�� and analyzes the jump conditions atthe boundary of each component to deduce that � takes only two values andsolves �in the distributional sense� a wave equation with characteristics n�

��

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and n�� Using more subtle arguments Kirchheim �Ki �� has recently proveda similar rigidity result for

K � SO���U� � SO���U� � SO���U�

where U� � diag��� �� �� and U�� U� are obtained by permuting the en�tries� An important additional di�culty is that SO��� is not abelian andone can thus not derive a linear equation for a suitable lift like � in thetwo�dimensional case�

� Length scales surface energy and singular perturba�

tions

Minimization of elastic energy does not determine the length scale of themicrostructure� If the in�mum of the energy is not attained� minimizing se�quences develop increasingly �ner oscillations and generalized Young measuresolutions correspond to an �in�nitely �ne microstructure� Also the Youngmeasure only records the asymptotic volume fraction of each phase but nottheir geometric arrangement�It is believed that addition of small surface �or higher gradient� energy contri�bution can remedy these de�ciencies� The precise form of the surface energyterm is not obvious� Popular choices for the total energy are

I��u� �

Z

W �Du� � ��jD�uj�dx

or

I��u� �

Z

W �Du�dx�

Z

�jD�uj�

where jD�uj is understood as a Radon measure� The idea is to study mini�mizers of these functional in the limit �� �� Almost nothing is known aboutthe three�dimensional model with full SO��� symmetry but results on lowerdimensional models already show very interesting behaviour�

��� A one dimensional model

As a caricature of the two�gradient problem consider

��

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I�u� ��R�

�u�x � ��� � u�dx�� min ����

subject to periodic boundary conditions� Clearly I�u� � � since the condi�tions u � � a�e� and ux � �� a�e� are incompatible� On the other handinf I � �� since a sequence of �nely oscillating of sawtooth functions uj canachieve ujx � f��g� uj � � uniformly� For any such sequence ujx generatesthe �unique� Young measure � � �

��� �

���� Note that there are many �dif�

ferent sequences that generate this Young measure�Minimizers of the singularly perturbed functional

I��u� �

�Z�

��u�xx � �u�x � ��� � u�dx

yield a very special minimizing sequence for I�

Theorem ��� ��Mu ���� If � � � is su�ciently small then every minimizerof I� �subject to periodic boundary conditions� is periodic with minimal periodP � � �������� �O�������

��� Surface energy and domain branching

Consider the two�dimensional scalar model problem �see �KM ��� for therelation with three�dimensional elasticity�

I�u� �

�Z�

LZ�

u�x � �u�y � ���dx dy�� min

u � � on x � �� ����

The integrand is minimized at Du � �ux� uy� � ������� The preferredgradients are incompatible with the boundary condition� The in�mum of Isubject to ���� is zero but not attained� The gradients Duj of any mini�mizing sequence generate the Young measure �

������ �

������� One possible

construction of a minimizing sequence is as follows �see Figure ��� Let sh be aperiodic sawtooth function with period h and slope�� and let u�x� y� � sh�y�for x � � u�x� y� � x

�sh�y� for � � x � Then consider a limit h� �� � �

such that h remains bounded� Similar reasoning applies if we replace ����

��

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δ L

u=0

u =-1 yu =1 y

h

h/2linear interpolation

Figure �� Construction of a minimizing sequence�

by the condition that u vanishes on the whole boundary of ��� L�� ��� ���

To understand the in�uence of regularizing terms on the length scale andthe geometry of the �ne scale structure we consider

I��u� �

�Z�

LZ�

u�x � �u�y � ��� � ��u�yydx dy�

subject to ����� Instead of the second derivatives in y one can consider otherregularizing terms� e�g� jD�uj�� The derivatives in y are� however� the mostimportant ones� since we expect that �ne scale oscillations arise mainly inthe y direction� It was widely believed that for small � � � the minimizersof I� look roughly like the construction uh�� depicted in Figure � �with thecorners of the sawtooth �rounded o� and optimal choices ���� h����� Thisis false� Indeed a short calculation shows that ��� � ��L����� h��� � ��L����

and I��u�h�L� � ����L���� On the other hand one has

Theorem ��� ��Sch ����� For � � � there exists constants c� C � �such that

c����L��� � minu�� at x��

I� � C����L����

The upper bound is obtained by a smooth version of the self�similar con�struction depicted in Figure �The mathematical issues become clearer if we replace I� by a sharp interface

version

��

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L

interpolation } h ~ ε L1/3 2/3

2θ θL L

Figure � The self�similar construction with � � � � �� Only twogenerations of re�nement are shown�

J ��u� �LR�

�R�

u�x � �juyyjdydx ����

subject tojuyj � � a�e� ����

Thus y �� u�x� y� is a sawtooth function andR �

�juyyjdy denotes twice the

number of jumps of uy� Minimization of ���� subject to ���� is in fact apurely geometric problem for the set

E � f�x� y� � uy�x� y� � �g�

The �rst term in J � is a nonlocal energy in terms of E� while the second isessentially the length of �E �more precisely its projection to the x�axis� asbefore we consider this to be the essential part since oscillations occur mainlyin the y direction�� The functional and the constraint are invariant underthe scaling

u��x� y� � ���u�����x� �y�

which suggests a self�similar construction with � ����

�����

Theorem ��� ��KM ����� For � � �

c����L��� � min��������

J � � C����L����

��

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Moreover if !u is a minimizer of J � subject to ����� ����� then

c����l��� ��R�

lR�

!u�x � �j!uyyj � C����l���� ����

The scaling in ���� is exactly the scaling predicted by the self�similar con�struction with � � ��

������

Conjecture ��� As �� �� l� � the rescaled minimizers

v��x� y� � �����l����!u� �lx� ����l���y�

converge to a self�similar function �at least for a subsequence��

The prediction of re�nement of the microstructure �domain branching� to�wards the boundary x � � in the simple models ����� ���� inspired newexperimental investigations ��Sch ����� In closely related models for mag�netization domains in ferromagnetic materials domain branching is experi�mentally well established ��Li ���� �Hu ��� �Pr ���� a rigorous mathematicalanalysis is just beginning to emerge� Already a quick look at some of thesophisticated constructions in �Pr �� suggests that a lot is to be discovered�

Outlook

To close I brie�y mention three questions in the analysis of microstructuresthat are widely open�� Which mathematical object describes microstructure most e�ciently"� What is the r#ole of dynamics"� How can one e�ciently compute microstructures"

��� Beyond Young measures

Young measures are but one way to extract �relevant information from arapidly oscillating sequence� They describe the asymptotic local phase pro�portions in a �ner and �ner mixture� The Young measure contains no infor�mation about the geometry of the mixture� self correlations of the sequenceor relevant length scales�

There is an intense search for new objects that record more information�and Tartar s article �Ta ���� from which the title of this subsection was bor�rowed� gives a recent survey� Due to space constraints I can only brie�y

��

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mention two such objects� the H�measure and its variants which alreadyfound many applications and the two�scale Young measure which still has toprove its usefulness�

A typical example where knowledge of local phase proportions is insu�cientarises in the theory of homogenization� A �ne laminate of a good and a poorheat conductor behaves macroscopically like an anisotropic material thatconducts poorly in the direction of lamination and well in the perpendiculardirections�To take the in�uence of the layering direction into account Tartar �Ta ���

introduced the H�measure that acts simultaneously on real space and onFourier space� The same object was introduced independently by G�erard�Ge ��� under the name �microlocal defect measure � For every sequenceuj � � L���� there exists a subsequence ujk and a Radon measure � on!� � Sn�� �the H�measure of fujkg� such that for every pseudo�di�erentialoperator A of order zero with �su�ciently regular� symbol a�x� �� one has

hAujk� ujkiL� �

Z��Sn��

ad��

For Rm�valued sequences one similarly obtains a matrix�valued �hermitian�measure �� ��ij���i�j�m� Applications of H�measures include small ampli�tude homogenization� compensated compactness with variable coe�cients�compactness by averaging in kinetic equations and the propagation of en�ergy concentrations in hyperbolic systems�Comparing H�measures and Young measures one sees that the former pre�

dicts limits that involve pseudo�di�erential operators while the latter onlyyields limits of local expressions �cf� ������� On the other hand the use ofH�measures is restricted to quadratic expressions while Young measures canhandle arbitrary nonlinearities� A synthesis of the two concepts is a majorchallenge for the future� So far even a good theory for a trilinear analogueof the H�measure is outstanding� Also the relation between H�measures andYoung measures is not known �partial results were obtained by Murat andTartar �MT ��� �Ta �����

As regards length scales� G�erard �Ge ��� introduced a variant of the H�measure� called semiclassical measure� that allows one to study oscillationswith a typical length scale hj � �� A similar measure was later considered

��

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by Lions and Paul �LP ����A di�erent approach to the resolution of length scales is the notion of

two�scale Young measures or Young measures on patterns ��AM ���� Forillustration consider a sequence of periodic functions wj � ��� ��� R that isuniformly bounded� jwjj � M � For a given sequence of scales hj � � onede�nes a new sequence

vj�x� y� � wj�x � hjy��

For L � � the set K � fg � L���L� L� � jgj � Mg is a compact subsetof L���L� L� equipped with the weak� topology and vj can be viewed as amap

Vj � ��� �� � Kx �� vj�x� ��

Since K is a compact metric space �a subsequence of� the sequence Vj gen�erates a Young measure � � ��� �� � M�K�� This Young measure �calledthe two�scale Young measure of the original sequence wj� is a probabilitymeasure on functions� obtained by blowing up the scale hj� Using this ideaone can describe the asymptotic behaviour as �� � of minimizers of

$I��u� �

�Z�

��u�xx �W �ux� � a�x�u��

where � c � a�x� � C �cf� section ���� If u�j is a sequence of minimizers�

�j � �� wj�x� � �����j u�j�x�� hj � �

���j and � is the Young measure generated

by Vj then ��x� is supported on �translates of� sawtooth functions withperiod ���a�x����� �cf� Theorem ���� Two�scale Young measures generalizethe concept of two�scale convergence ��Ng ��� �Al ���� which is very useful ifthe sequences involved are well approximated by functions that are periodic�with �xed period� in the fast variable�

��� Dynamics

So far we have only considered time independent situations and studied min�imizers or almost minimizers of the energy� A typical justi�cation is thatthe time dependent problem admits a natural Liapunov function �free en�ergy� entropy� ���� and therefore the evolution of �generic initial data shouldconverge to �at least local� minimizers of the Liapunov functional�

��

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Is such reasoning still reasonable if minimizers of the Liapunov functionaldo not exist and minimizing sequences have to develop microstructure"Friesecke �Fr ��� and Friesecke and McLeod �FM ��� studied a one dimen�sional model problem �viscoelastic bar on a foundation� whose Liapunovfunctional is essentially given by ���� �strong convergence of the kinetic en�ergy is easy�� They solved a long standing conjecture by showing that thedynamics excludes formation of �ner and �ner microstructure and hence en�ergy minimization� See Ball et al� �BHJPS ��� and Pego �Pe � for relatedmodels and Ball �Ba ��� for a general discussion of energy minimization ver�sus dynamics� Virtually nothing is known for the long�time dynamics ofhigher dimensional models� despite the interesting numerical simulations ofSwart �Sw ����

Another important issue is the evolution of microstructures� If the mi�crostructure at each point in time is described by a Young measure �or someof the objects discussed in section ��� is it possible to deduce an evolu�tion law for the Young measure" Conceptually this is similar to the closureproblem in turbulence models and one might thus be pessimistic� Nonthelessthere has been interesting progress over the last years �see �KP ���� �FBS �����HR ���� �De ���� and in particular recent work of Otto �Ot ���� �Ot ���� Aninteresting new approach to dynamics that takes into account the presenceof many shallow local minima apperas in �ACJ ����

��� Computation

Computation of microstructures through numerical energy minimization isa very challenging task� There are presumably many local minimizers forthe discrete and the continuous problem� the Euler�Lagrange equations aremixed elliptic�hyperbolic and oscillations typically develop on the scale ofthe discretization� which renders results very discretization dependent� Fora recent review of the state of the art see Luskin �Lu ��� and the referencestherein� So far� most numerical methods make no use of available analyticalinformation such as the classi�cation of gradient Young measures� the mi�nors relations or algorithms for rank�� convexi�cation �see sections � and of�Lu ��� for the discussion of some exceptions�� It seems to me that an impor�tant question is how to represent microstructures numerically in an e�cientway� Ideally a good representation should both yield a high compression rateand be adapted to the numerical algorithm� This issue is closely related to

Page 30: MPI MIS Leipzig, Preprint Nr. 3

the search for better analytical descriptions of microstructure �see section ��above��

Acknowledgements

The writing of this article and the work reported here would not have beenpossible without the support and encouragement of many colleagues andfriends� Particular thanks go to G� Alberti� J�M� Ball� K� Bhattacharya�G� Dolzmann� I� Fonseca� G� Friesecke� R�D� James� D� Kinderlehrer� B� Kirch�heim� J� Kristensen� R�V� Kohn� V� �Sver�ak and L� Tartar�

Page 31: MPI MIS Leipzig, Preprint Nr. 3

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�Ed ��� R�E� Edwards� Functional analysis� Holt� Rinehart and Win�ston� �����

�Er �� J�L� Ericksen� Some phase transitions in elastic crystals� Arch�Rat� Mech� Anal� �� ������ �� ����

�Er �� J�L� Ericksen� The Cauchy and Born hypothesis for crystals�in� Phase transformations and material instabilities in solids�M�E� Gurtin� ed��� pp� �� � Academic Press� ����

�Er �� J�L� Ericksen� Weak martensitic transformations in Bravais lat�tices� Arch� Rat� Mech� Anal� �� ������ �� ���

�Fo �� I� Fonseca� Phase transitions for elastic solid materials� Arch�Rat� Mech� Anal� �� ������ ��� ����

�FBS ��� I� Fonseca� D� Brandon and P� Swart� Dynamics and oscillatorymicrostructure in a model of displacive phase transformations�in� Progress in partial di�erential equations� the Metz surveys�� �M� Chipot� ed��� Pitman� ����� pp� ��� ����

�FMP �� I� Fonseca� S� M�uller and P� Pedregal� Analysis of concentra�tion and oscillation e�ects generated by gradients� to appearin SIAM J� Math� Anal�

�Fr ��� G� Friesecke� Static and dynamic problems in nonlinear me�chanics� Ph�D� thesis� Heriot�Watt University� Edinburgh������

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�FM ��� G� Friesecke and J�B� McLeod� Dynamics as a mechanism pre�venting the formation of �ner and �ner microstructure� Arch�Rat� Mech� Anal� ��� ������� ��� ���

�Ge ��� P� G�erard� Mesures semi�classique et ondes de Bloch� in� Equa�tions aux d�eriv�ees partielles� expos�e XVI� seminaire ���� ���Ecole Polytechnique� Palaiseau�

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�Gr �� M� Gromov� Partial di�erential relations� Springer� ����

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�Hu �� A� Hubert� Zur Theorie der zweiphasigen Dom�anenstrukturenin Supraleitern und Ferromagneten� Phys� Status Solidi �������� ��� ���

�IT ��� A� ' C� Ionescu Tulcea� Topics in the theory of liftings�Springer� �����

�Kh �� A� Khachaturyan� Some questions concerning the theory ofphase transformations in solids� Soviet Physics�Solid State ������� ���� ����

�Kh �� A� Khachaturyan� Theory of structural transformations insolids� Wiley� ����

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�Ki � D� Kinderlehrer� Remarks about equilibrium con�gurationsof crystals� in Material instabilities in continuum mechanics�Heriot�Watt Symposium� �J�M� Ball� ed��� Oxford UP� ���pp� �� ����

�KP ��� D� Kinderlehrer and P� Pedregal� Characterization of Youngmesures generated by gradients� Arch� Rat� Mech� Anal� ���������� ��� ����

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�KP ��� D� Kinderlehrer and P� Pedregal� Gradient Young measure gen�erated by sequences in Sobolev spaces� J� Geom� Analysis �������� �� ���

�Ki �� B� Kirchheim� in preparation�

�Ko �� R�V� Kohn� The relationship between linear and nonlinear vari�ational models of coherent phase transitions� in� Trans� �thArmy Conf� on applied mathematics and computing� �F� Dres�sel� ed��� ����

�KM ��� R�V� Kohn and S� M�uller� Branching of twins near anaustenite(twinned�martensite interface� Phil� Mag� A ��

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�KM ��� R�V� Kohn and S� M�uller� Surface energy and microstructurein coherent phase transitions� Comm� Pure Appl� Math� ��������� ��� ����

�Kr ��� J� Kristensen� Finite functionals and Young measures gener�ated by gradients of Sobolev functions� Ph�D� Thesis� TechnicalUniversity of Denmark� Lyngby�

�Kr �� J� Kristensen� On the non�locality of quasiconvexity� to appearin Ann� IHP Anal� non lin�eaire�

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�Lu ��� M� Luskin� On the computation of crystalline microstructure�to appear in Acta Num�

�Ma ��� J�P� Matos� Young measures� regularity results in the two wellproblem� in� Int� Conf� on Di�erential Equations� EQUADIFF��� Barcelona ����� �C� Perello� C� Simo� J� Sola�Morales� eds���World Sci� Publ�� ����� pp� �� ���

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�Mo ��� C�B� Morrey� Quasi�convexity and the lower semicontinuity ofmultiple integrals� Paci�c J� Math� � ������� �� ���

�Mo ��� C�B� Morrey� Multiple integrals in the calculus of variations�Springer� �����

�Mu ��� S� M�uller� Singular perturbations as a selection criterion forperiodic minimizing sequences� Calc� Var� � ������� ��� ����

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�Mu �c� S� M�uller� A sharp version of Zhang s theorem� manuscript�

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�MS �� S� M�uller and V� �Sver�ak� in preparation�

�MT �� F� Murat and L� Tartar� On the relation between Young mea�sures and H�measures� in preparation�

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�Ng �� G� Nguetseng� A general convergence result for a functionalrelated to the theory of homogenization� SIAM J� Math� Anal�� ������ �� ����

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