Parametrized Measures and Variational Principles

+ (3. (8 -

t),

8 E K.

In particular for J-t-a.e. x E 0,


+ (3. (J(x) -

t),

and integrating over 0" we get (1-12). (Keep in mind that J-t(o')

= 1.)

•

We also remember in this section a few general facts about weak convergence in LP(O,). Let 1 ::::: p < 00. We say that {fJ} converges weakly to f in LP(o') and write fJ ~ f in LP(O,) if

In fJ(x)g(x) dx In f(x)g(x) dx, ----t

for every 9 E Lq(O,), l/p+ l/q = 1. For P = 00, {fJ} converges weakly in LOO(O,) (in written form fJ . . ":.,. f in LOO(O,)) if

* to f

In fJ(x)g(x) dx In f(x)g(x) dx, ----t

for every 9 E L1(0,). The criterion for weak compactness is contained in the following proposition. The case p = 1 is very special. Proposition 1.3 Let 1 < p ::::: 00. The sequence {fi} is weakly relatively compact in LP (0,) (weakly * relatively compact if p = 00) if and only if there exists a constant K ? 0 such that IlfJIILP(n) : : : K uniformly for all j. Let p = 1. The sequence {fJ} is relatively compact in L1 (0,) if and only if: i) there exists a constant K ? 0 such that IlfJllu(n) : : : K for all j; ii) for every E > 0 there exists a 8 = 8(E) > 0 such that for every measurable subset E with lEI < 8, we have

LIfJ(x)1

uniformly for all j.

dx <

E,

23

7. Bibliographical remarks

Condition ii) is the equiintegrability property or Dunford-Pettis criterion of weak compactness in L 1 (r2). It is a well-established fact that L 1 (r2) is very peculiar from the point of view of weak convergence and that peculiarity is precisely condition ii) above. Concentrations effects are connected to the failure of ii). Proposition 1.4 and only if

Let 1 < p

:s;

00.

fj

~

f in U(r2) (or weak

* in

Loo(r2)) if

i) Ilfj Ib(S"l) :s; K, K> 0; ii) limj->oc J[)(fj(;x;) - f(x)) dx = 0, for all cubes Dc r2.

Let p = 1. h ~ f in L1 (r2) if and only if i) IlfiIILl(n) :s; K, K > 0: ii) the equiintegrability property holds; iii) limi->oo Jo(jj(x) - f(:r))dx = 0, for all cubes Dc r2.

These notions of weak convergence in the LP(n)-spaces can be translated to the Sobolev spaces W1,p(n). As usual we take

Weak convergence in W1,p(n) means weak convergence in LP(r2) for the functions and their gradients. 7. Bibliographical remarks

Many textbooks dealing with the calculus of variations are available. Some of them are less advanced ([7], [310]), broader in scope ([41], [70], [153], [157], [294]) or follow a more classical approach ([89], [158], [309]) while others are based on the direct method and on weak convergence techniques ([61]' [98], [239], [313]' [316]). An important complement on variational methods is convex analysis. We have included two basic such references [118], [274]. There are also many hooks on weak convergence. Most of them are graduate texts in functional analysis or partial differential equations. [6], [56], [115], [116], [132]' [160], [284] are basic references. In the context of parametrized measures, where some of the examples in Section 1 can be found, [302] must be looked at. As an introduction to weak convergence, nonlinear functionals and parametrized measures the reader may try [93] as well. The subject of parametrized measures is not new, especially for optimal control experts. There are a number of works on parametrized measures treated from a rather general and (for some of them) abstract point of view. These include [24], [27], [212], [213], [282]' [314], [315] and the ones already cited. Most of the ideas related to compensated compactness and the general framework described in Section 2 are scattered through the literature. Some of the fundamental sources for compensated compactness and the theory of homogenization are [247], [248], [249], [250], [303], [304], [306]. Applications

24

Chapter 1. Introduction

of these methods to partial differential equations have been quite successful. In addition to the previous references, the reader is referred to [110], [111], [112], [113], [114], [265], [277], [286], [288]. These are included here as a sample since there exists a very large amount of work dealing with techniques in compensated compactness and homogenization. A nice account of the use of weak convergence in nonlinear partial differential equations is [132]. Parametrized measures have been used in evolution problems as well. Some examples in this direction are contained in [107], [149], [150], [178], [190], [290]. One of the main drawbacks of parametrized measures is that since they keep track of statistical properties alone, they are not well suited for problems in mathematical physics in which transport properties, multiscale phenomena, or more precise information concerning oscillations, playa fundamental role. To solve these difficulties H-measures have been introduced in [154], [155], [305]. Another point is that parametrized mea.<;ures have not been designed to detect concentrations since they completely miss this effect on sequences. Several references concentrate on this issue [9], [145], [203], [204]. Additional references on compensated compactness and parametrized measures related to variational principles are [79], [80], [142], [255], [258]. There are topics beyond the scope of this book that are important from the point of view of variational principles, integral representations, functions of bounded variations, r-convergence, etc. A few basic references for these subjects are [18], [103], [161]. The proof of the classical Jensen's inequality for convex functions and probability measures included here is the one in the excellent book [283]. Other references on measure theory, functional analysis, partial differential equations, etc, where background material can be studied are [47], [90], [135], [164], [214], [268], in addition to the ones mentioned in the second paragraph of this section. Most of the references on parametrized measures call them Young measures after the pioneering work of Young. We take the two terms as equivalent in this text.

Chapter 2 Some Variational Problems

1. Introduction

This chapter gathers a collection of problems for which the analysis does not involve any differential constraint, or if it does it is in a somewhat elementary way. It is a good way of practicing with the general ideas we will pursue for more complicated situations in subsequent chapters. For this reason we do not pretend to give the sharpest hypotheses under which theorems can be proved or improved, but rather focus on understanding the main techniques in each example. Some formal proofs are left to the reader as exercises. The same principle explains why in some of the problems we do not pursue the proof of all the steps and lemmas used when they are not relevant to our discussion. Three of the examples refer to variational principles or optimization. The last one does not. This has been included with the sole purpose of providing an illustration of how some analysis in terms of parametrized measures can also be helpful and provide some insight even though the problem is not directly related to variational principles but it is placed in a completely different context: large time behavior of complicated turbulent systems. The first example in Section 2 corresponds exactly to the case of no differential constraint in the context of Chapter 1, Section 3: .c is the set of bounded sequences in some LP(O). The associated notion of convexity is the usual convexity so that Jensen's inequality does not place any real restriction on families of probability measures. A complete proof of a characterization theorem for the case p = CXl is given. It is interesting to realize how differential constraints included in .c make the analysis so much harder: the entire Chapter 8 (based on Chapter 6) is devoted to the characterization of parametrized measures under the curl constraint. In Section 3, a short discussion of simple optimal control problems is considered. Existence theorems and relaxation are described and examined in terms of parametrized measures associated to pairs of controls and states. We restrict ourselves to control problems governed by ordinary differential equations. Hypotheses are not completely explicit again because we want to emphasize the underlying analysis. Finally, an optimal design problem is described and a solution given in the form of parametrized measures. We are P. Pedregal, Parametrized Measures and Variational Principles © Birkhäuser Verlag 1997

26

Chapter 2. Some Variational Problems

looking for the optimal design of a plate under the action of an external load. The optimality criterion is to minimize the compliance of the plate taken as the work done by the load. Minimizing sequences develop finer and finer oscillations but the weak limit of such sequences do not furnish the optimal design sought. Some generalized relaxed functional needs to be considered where families of probability measures compete. We will go back to this phenomenon of very fine spatial oscillations in the more involved applications of Chapter 5. Section 5 contains a short discussion of coherent structures and small-scale fluctuations in some turbulent systems. 2. Variational problems under no differential constraint This section aims to be an introduction to variational problems in which there is no derivative or differential constraint involved. The prototype of such problems is

I(u) =

In

cp(x, u(x)) dx,

(2-1)

where competing functions u : 0 C RN ~ Rm are vector valued functions in some V(O) space. 0 is assumed to be a regular domain and cp(x, >.) : RN X Rm ~ R* is some known integrand assumed to be measurable in x and continuous in >.. We further consider the typical volume constraint

In

u(x) dx

=

0:,

(2-2)

where 0: E RTn is given a priori. The most basic interpretation of this problem (for m = 1, the scalar case) consists in finding the best mass distribution (or the distribution of some other quantity), u, according to the optimization criterion of minimizing the integral (2-1) for a total amount of mass 0: given in (2-2). For the vector case m > 1 we are looking for the best distribution of several constituents for a fixed amount of each of them. Suppose we want to find optimal solutions (minimizers) of our problem (2-1) and (2-2) through the direct method as described in Chapter 1. We need to understand the conditions on the density cp that ensure the weak lower semicontinuity property in LP(O). Through coercivity assumptions on cp of the type (2-3) c(t>.t P -1):::; cp(x,>,), p> 1,c > 0, minimizing sequences will be uniformly bounded in LP(O). Consider £,

= {{ Uj}

and define for any function 'l/J : R Tn

bounded in P(O)} , ~

R *,

2. Variational problems under no differential constraint

27

If A = t.>'l + (1 - t)A2' after the examples in Chapter 1, it should be clear that we can find a bounded sequence in Loo(D), {Uj}, whose parametrized measure is precisely 1J = tl5)'l + (1 - t)I5 A2 , homogeneous. Indeed we can take Uj(x)

= A2 + Xt(jx· n) (AI - A2)

for any vector n E R N , where Xt is the characteristic function of (0, t) in (0,1) extended by periodicity. Therefore

Likewise, by induction, if

there exists a bounded sequence in L 00 (n), {Uj}, whose corresponding parametrized measure is

and thus

By taking the infimum in the right-hand side we arrive at

where 'lj;** is the usual convexification of 'lj;. The reverse inequality is also true. By Jensen's inequality

I~I

L

'lj;(Uj(x)) dx

~ I~I

L

'lj;**(Uj(x)) dx

~ 'lj;** (I~I

L

Uj(X)dX).

The limit of the right-hand side is 'lj;** (A) for any sequence {Uj} converging weakly to A. In this case 'ljyC = 'lj;** and all the required assumptions in the framework described in Chapter 1, Section 3 are trivially true since Jensen's inequality with respect to all convex functions does not place any restriction on families of probability measures. Theorem 2.1 Let 1J = {lJ x } xEO be a family of probability measures depending measurably on x E n such that supp (lJx ) eKe R m , a.e. x E n. There exists a sequence, {u.d, bounded in LP(n) and taking values in K, whose parametrized measure is 1J if and only if

r r IAI

for p <

iniK 00,

P

dvx (A) dx <

00,

and if and only if K is bounded for p =

00.

(2-4)

28

Chapter 2. Some Variational Problems

Proof We provide the details for the case p = 00 in which we assume that K is bounded. The extension to the finite case p < 00 requires approximation and truncation techniques that are not relevant to the main stream of our discussion. They will be treated in Chapter 7. If there exists a sequence, {Uj}, bounded in LP(n), the associated parametrized measure satisfies (2-4), by Theorem 6.11. The restriction on the support of each Vx is also elementary if the image for each Uj lies inside K. The converse is the interesting part of the theorem. We place ourselves in the Banach space M (D x K) of Radon measures on D x K under the total variation norm, where we assume n to be bounded and K compact, so that the product D x K is compact. In this case this Banach space is the dual of C(D x K) endowed with the sup norm which is separable. Convergence in M(D x K) with respect to the weak * topology can be characterized by sequences. Consider

A = {fLU E M(D x K) : (fLu, 1/J) = l1/J(x, u(x)) dx,

1/J

E

C(D x K), U

:

n -+ K} ,

and let fL E M (D x K) be defined by

(/L,1/J)

=

1~ll

L

VJ(x,.\) dvx (.\) dx,

where the family v = {vXLEO is given verifying supp (v x ) C K a.e. x E n. For the case p = 00, (2-4) drops out and is replaced by the boundedness restriction onK. Step 1. A is a convex set (closure is meant in the weak * topology). Let t E (0,1) and let U;, i = 1,2 be measurable functions taking values in K. We would like to show that the measure tfLul + (1 - t)fL U2 belongs to A. Let once again Xt stand for the characteristic function of (0, t) in (0,1) extended by periodicity to all of R, and write Xk(X) = Xt(kx . n) for any non-zero vector n ERN. As we have argued in Chapter 1, Xk ~ t in LaO (n). Consider

We claim that fLu(k) ~ tfLul + (1-t)fLu2 in M(D x K). In fact for any continuous

1/J,

lim

k--+oo

J

1/J(x, u(k) (x)) dx = lim

r1/J (x, Xk(X)Ul (x) + (1 - Xk(X)) U2(X)) dx r [Xk(X)1/J(X,Ul(X))

k--+oo } 0

=

lim

k--+oo } II

+ (1 -

Xk(X)) 1/J(x, U2(X))] dx =(tfLul +(1-t)fLu2,1/J)·

3. Optimal control problems

29

Step 2. pEA. For this we use the Hahn-Banach theorem. Suppose T is a linear functional represented by a particular function 'IjJ E C(0 X K) such that \Pu, 'IjJ) 2: 0 for all measurable U : n - t K:

10 'IjJ(x,u(x))dx 2: o. Take u(x)

= min\ 'IjJ(x, >.).

r

JnxK

Since 'IjJ(x, >.) 2: 'ljJ(x, u(x)) and p is non-negative,

'ljJ(x, >.)dp(x, >.) 2:

=

r

JnXK

'ljJ(x,u(x))dp(x,>.)

10 'IjJ(x, u(x)) dx

2: 0, by (2-5). Step 3. Conclusion. Because of the remark made earlier about how sequences characterize weak * convergence in M (r"2 x K), by Step 2 we can find a sequence of measurable functions, {Uj}, such that lim J--+OO

Inr 'IjJ(x,uj(x))dx = Inr JrK 'IjJ(x,>')dvx(>')dx,

for all continuous 'ljJ. In particular, if'P : K - t R is continuous the sequence (or some suitable subsequence) {'P( Uj)} will converge, weakly * in the sense of measures, to

By uniqueness of the limit 'P(Uj) ...=.. VJ in LOO(o.) so that v = {vXLEn is the parametrized measure associated to {Uj}. • Based on this characterization, we can proceed to analyze weak lower semicontinuity and relaxation for I in (2-1) along the lines developed in Chapter 1, Section 3. The reader is invited to provide the details. Notice that the constraint on the total mass (2-2) is preserved under weak convergence.

3. Optimal control problems Optimal control is a part of the theory of optimization more general than the calculus of variations. We would like to study as an example one of the most basic optimal control problems governed by ordinary differential equations in order to show how parametrized measures may serve to analyze this type of problems as well. As a matter of fact, our general framework in Chapter 1 is also useful in this context.

30

Chapter 2. Some Variational Problems Our (payoff) functional I is of the form

I(u,y)

=

i

'P(t,u(t),y(t))dt

where J is some interval (finite of infinite) of R, u : J -> Rm is the control variable (the free variable) and y : J -> Rd is the state of the particular system under consideration coupled to the control through the equation of state

y'(t) = A(t, u(t), y(t)), where A : J x Rm x Rd -> Rd is such that existence of solutions to the equation of state are guaranteed for the class of controls we want to consider. There might be some other constraints in the problem like u(t) E K for some fixed subset K c R m or restrictions on initial conditions for the equation of state. For definiteness we neglect these other conditions, or assume them to be preserved by weak convergence otherwise. This last hypothesis might not be true, though, in some circumstances of interest and may require some further analysis. Assume that 'P : J x Rm x Rd -> R is continuous in all its arguments and we have the coerciveness hypothesis

c(lul P+ lylP -

1) :S 'P(t, u, y),

p> 1, c > 0,

for all (t,u,y) E J x Rm x Rd. Minimizing sequences will be bounded in LP(J) under this assumption. We consider

Assume further that

IA(t, u, y)1 :S C (Iul q+ Iylq + 1), In this case, if {(Uj, yj)} E

£, and /.l = {/.It

parametrized measure, then {IA(t,uj,Yj)I

hEJ

P/ q }

1:S q < p. is the associated underlying

is bounded, and since q

< p,

{A( t, Uj, yj)} (or some appropriate subsequence) converges weakly in L1 (J). Consequently,

where A E Ll(J). If we define

:~.

Optimal control problems

31

modulo a constant, Yj -+ Y strong in Loo(J). Recalling the comments about how strong convergence is reflected on the parametrized measure, Proposition 6.13, we conclude that if v = {vt} tE.! is the parametrized measure associated to the sequence of controls {11)} then

ILt = Vt and

Q9

Oy(t),

a.e. t E J,

(2-6)

A(t) = lmA(t, A, y(t)) dVL()\)'

Sequences in £ correspond to strongly convergent sequences in L 00 (.1) for the state variable and weakly convergent sequences in LP (J) for the control variable. This in particular implies that the appropriate notion of convexity (associated to £) for weak lower semi continuity is usual convexity of r.p with respect to the control variable: if 'I1j ~ u in LP(rl) and Yj -+ Y in L=(rl) then under convexity of r.p with respect to 1L and bounded ness from below, by Theorem 6.11, (26) and Jensen's inequality, we have lim

r r.p(t, lLj(t), 1Jj(t)) dt:.::: .JJr .JRm r

J---'>=.JJ

xRrl

r.p(t, Al, A2) dILt(Al, A2) dt

1lm r.p(t, A, y(t)) dVt(A) dt :.:::1 r.p (t, lm A dVt(A), y(t)) dt

=

=

1

r.p(t, u(l), y(t)) dt,

if a suitable subsequence has been chosen. This time, however, this condition does not ensure by itself the success of the direct method to achieve minimizers for our problem. Indeed, £ is not weakly closed. To see this, suppose {( Uj, YJ)} is a sequence in £ so that Yj -+ Y and Uj ~ u. The crucial question is whether 11 and yare coupled by the equation of state. We know that if v is the parametrized measure associated to the sequence of controls then

y'(t) =

lm A(t,

A, y(t)) dVt ()\),

so that for weak closed ness we must require

lm

A(t,A,y(t))dVt(A) = A(t,u(t),y(t)),

u(t) =

lm

Advt(A).

This condition does not hold for all choices of v unless A is linear in 11. In the spirit of the discussion of Chapter 1, the reader can rigorously prove the following existence theorem for the optimal control problem.

32

Chapter 2. Some Variational Problems

Theorem 2.2 Assume tbat tbe following bypotbeses bold: i) A(t,u,y) = A 1 (t,y)u+A 2 (t,y) wbereA 1 : JxR d ----; Mmxd, A 2 : JxRd----; Rd and

ii) r.p is continuous, convex in u and

c(lul P+ lylP -1)::::; r.p(t,u,y),

p> 1.

Tbe associated optimal control problem admits a solution.

If A is not linear in u, even though r.p may be convex on the control variable, the analysis might proceed seeking a relaxed or generalized functional defined on parametrized measures associated to sequences of controls 1(v)

=

rr

JJ JRrn

where y'(t) =

and v

= {VthEJ

r

JRrn

r.p(t,)..,y(t))dvt(>\)dt,

A(t,>.,y(t))dvt('>'),

satisfies

Again there might be more restrictions on 1/ coming from the additional initial constraints. There are, however, some technical difficulties to be overcome with this generalized formulation related to the differential equation for y which is written this time in terms of a family of probability measures. 4. An optimal design problem

We describe in this section some analysis of an optimal design problem for a plate of variable thickness under the model of Kirchhoff for pure bending of symmetric plates. We try to find the optimal structure with respect to the overall rigidity of the plate under the action of an external load. The model we consider is a somewhat simplified version where the thickness of the plate depends on just one variable and the tensors involved in the analysis depend upon the design of the plate through the half-thickness h. Let n be a regular, smooth domain in R2 representing the midplane of the plate with respect to which the plate is symmetric. The deflection or vertical displacement w in the model under consideration obeys the fourth order, elliptic equation

(2-7)

4. An optimal design problem

33

where FE L2(0) is the vertical load on the plate. The summation convention is used throughout this section. This equation must be satisfied in O. The design of the plate is hidden in the tensor M a {3,fJ through the dependence

where h is the thickness and B a {3,fJ is a constant tensor that depends on material constants alone. In order to use Lemma 2.3 below, we have to restrict ourselves to the case where the thickness h is in fact a function of Xl alone (though we will still write h(x), X EO), and Xl belongs to the interval

(a, b) = {Xl E R: there exists some

X2

E R with (Xl, X2) EO} .

We further restrict the class of materials by imposing a orthotropic condition: the nonzero components of B a {3,fJ are Bllll = B2222 = B1l22

B1212

=

B1221

= B22ll = =

=

B2ll2

E -1--2 ' -r Er -1--2 ' -r

B2121

E

= 2(1 + r)'

where E and r stand for the Young's modulus and the Poisson ratio, respectively. Equation (2-7) is completed with the boundary conditions

oW =0 an

W= -

'

on

an,

(2-8)

reflecting the hypothesis that the plate is clamped. The boundary value problem (2-7) together with the boundary conditions (2-8) is variational, so that the solution is indeed the minimizer of the functional

(again the summation convention is assumed) over Hg(O), the subspace of H2(0) satisfying (2-8). This can be easily checked. H2(0) is the Hilbert space of L2(0)-functions having first and second weak derivatives in L2(0). The compliance of the plate is defined to be the work done by the load F and is regarded as a function of the half-thickness h,

L(h)

=

In

Fwdx.

(2-9)

34

Chapter 2. Some Variational Problems

It yields a measure of the rigidity or flexibility of the plate under the action of F. The design or optimization object is to minimize L(h) among all the admissible plates with prescribed volume. The technical assumptions on the half-thicknesess h that may compete in (2-9) are the following

'}-{ = { hE Loo(n) : hmin :S:

h(x) :S: hmax ,

in

h(x) dx

= Vo } ,

where h min , h max and Vo are prescribed a priori in a consistent way

o < hmin Inl < Vo < hmax Inl· The basic feature of this optimization problem is the lack of minimizers. Minimizing sequences oscillate abruptly seeking the minimum value of the compliance available. In such cases a relaxation of the problem should be performed. What this amounts to is to provide some precise description, as simple as possible, of minimizing sequences. There might be many different types of minimizing sequences that realize the infimum of the compliance, some of them extremely complicated. To determine a relaxation is to search for a way to describe minimizing sequences with as few variables as possible. This description should be valid for all choices of the different parameters of the problem. The basic tool to describe relaxation in this context is the following wellknown lemma. It also explains why certain expressions (the cubic-average and harmonic cubic-average) arise in these relaxations. In order to state this result, we need some notation. A fourth order tensor M(x) is said to be orthotropic if the non-vanishing coefficients are M l l l l , M2222 and

M is bounded by the constants (d, D) if for every symmetric tensor t = have for every x E n

to;{3

we

d Itl 2

:S: Mo;{3,,/oto;{3t"/o, IMo;{3,,/oto;{3 I :S: D It I for every ,,/,8. Lemma 2.3 Let {Mk} be a sequence of orthotropic tensors bounded uniformly by (d, D). Let us assume that

k )-1 * (MOO )-1 (M1111 1111 --->.

,

(Mf122) (Mf111) -1 ~ (MU22) (MU1 d- 1 ,

(M~222) - (Mf122)2 (Mf111) -1 ~ (M~22) - (MU22)2 (MU11 )-l , k * MOO M 1212 1212· --->.

If w k , 1 :S: k ::; 00, is the solution of (2-7) and (2-8) corresponding to Mk, then wk --->. WOO in H5(n).

4. An optimal design problem

35

We examine relaxation directly in terms of parametrized measures and find easily a generalized minimizer. Once we achieve the existence of minimizers it is interesting to look for other minimizers, having in mind to simplify the understanding of minimizing sequences that generate such minimizers. In this sense, the motivation is to use as few design variables as possible to describe generalized minimizers. It should be noted that this process can be accomplished with this particular problem because the generalized compliance functional depends only upon certain moments of the parametrized measures associated to minimizing sequences. Let H be the set of parametrized measures associated to sequences hk of half-thicknesses. In view of Theorem 2.1, the only restriction we have on such families is the support and the volume integral H

= {fl = {tLx} xEO

:

supp flx C Q = [hmin' h max ] a.e. x E 11,

j ..J/").. dILx(>. ) dx = Vo} . n

Q

Notice that for any such fl we can find, according to Theorem 2.1, a sequence {hk} taking values in Q and whose associated parametrized measure is precisely fl. It might not be true. however. that

L

hk(x) dx = Vo,

for all k.

What we do know is that

To solve this problem is a pure technicality and involves changing each hk in a small set without changing the parametrized measure. The reader is invited to provide the details. See Lemma 6.:3 in Chapter 6 (this lemma has not been included in Chapter 1). In order to define a compliance in H, let us further examine Lemma 2.3. The different weak limits we should care about in our case in order to apply the lemma are

36

Chapter 2. Some Variational Problems

These weak limits can be represented through the moments of order 3 and -3 of the parametrized measure J-t corresponding to the sequence {h k }. Hence, if we let

m(x) c- 1 (x)

= ~ A3 dJ-tx(A). =

(2-10)

~ A-3 dJ-tx(A),

and define

(MITl1)-l

=

(~c(x) 1 ~r2)

-1,

(M~22) (MITll)-l = r,

(M~22) -

(MIT22)2 (MITll)-l =

(M~12) = ~m(x)

I!

~m(x)E,

(2-11)

r'

by Lemma 2.3 (the other hypotheses in the lemma are easily verified in this situation), the displacements Wk corresponding to the tensors Mk associated in turn to hk which generate J-t, will converge weakly to the solution of the same problem with the tensor Moo. Thus we must define the compliance L for elements in 'H to be

L(J-t)

=

In Fwdx,

where w is the solution of (2-7), (2-8), with the tensor MOO depending on J-t through (2-10) and (2-11).

Theorem 2.4

inf L = minL. 1-£ 'Fi

Proof At this point the proof of the theorem has almost been indicated. First, notice that for h E 'H, J-t = 8h (x) E 'H,

and, moreover, L(h) = L(J-t), so that infL < inf L. 'Fi - 1-£ On the other hand, given any J-t E 'H we can find a sequence {hd c 'H whose parametrized measure is J-t, as indicated. Again by Lemma 2.3 we conclude the weak convergence of the solutions to (2-7), (2-8) as before, and thus

The arbitrariness of J-t yields the equality of the two infima.

5. Turbulent fluids

37

To show existence of minimizers for (H, L) is now an easy task. Take any minimizing sequence for L in H. The parametrized measure generated by such sequence fJ is admissible since it belongs to H and by definition of L we have as before

•

so that fJ is truly a minimizer.

A crucial observation is that L depends only upon the moments of order 3 and -3 of fJ, in such a way that if fJ1 and fJ2 have in common these two moments then L(fJ 1 ) = L(fJ2). This brings us to the question of finding the easiest fJ E H that has the same moments of order 3 and -3 as a given minimizer fJ whose existence is guaranteed in Theorem 2.4. Let us set

m(x) =

c- 1(x)

=

10 >.3 dfJx(>'), 10 >.- 3dfJx(>'),

where fJ is a minimizer. Given Q, VO, m and c, the problem reduces to seeking a family of probability measures as simple as possible whose support is contained in Q, whose integral volume is Vo and whose moments of order 3 and -3 are m and c- 1 , respectively. Any family verifying these conditions is a minimizer for L and therefore any generating sequence of such a parametrized measure will be a minimizing sequence for our original optimization problem. Although it is beyond the purpose of this book, one can actually find minimizers for L of the form

for some O(x) E [0,1]' hE H, and>' E [a, b]. This generalized minimizer is the one that requires a minimal number of design variables. 5. Turbulent fluids

One of the most striking features of many turbulent fluid systems is the appearance of large-scale organized states, or coherent structures, in the midst of smallscale fluctuations. Such phenomena occurs, for example, in high Reynolds number two-dimensional hydrodynamics, and in slightly dissipative magnetofluids in two and three dimensions. The parametrized measure has proven to be a useful device in the modeling and analysis of coherent structures inherent in the long-evolved state of such systems. Roughly speaking, the parametrized measure represents a long-time weak limit of the relevant turbulent fluctuating fields, and the parametrized mean associated with this measure defines a

Chapter 2. Some Variational Problems

38

macroscopic organized state. Here, we illustrate these methods, focusing on two-dimensional hydrodynamics. The dynamics of an ideal, incompressible two-dimensional fluid is governed by the Euler equations:

OW

at

+u·\7w=o,

w(O,x)=wo(x).

(2-12)

Here u = (Ul' U2) is the fluid velocity and

is the scalar vorticity field. The equations are assumed to hold in a bounded, simply connected spatial domain D C R2 with smooth boundary aD. The velocity field is divergence free, \7 . u = 0, and tangential U . n = on aD. Consequently, there exists a stream function 7jJ(x) such that

°

U=

07jJ- -07jJ) (oX2'

°

oXl

'

with 7jJ = on aD. The stream function and vorticity are, therefore, related through the elliptic boundary value problem -6.7jJ

= w, in D,

7jJlan

= 0,

(2-13)

and thus the vorticity transport equation (2-12) can be expressed entirely in terms of w alone ow (2-14) +o(w,Gw) = 0,

at

where we have written 7jJ = Gw with G the Green's operator corresponding to the Dirichlet problem (2-13) and 0(1, g) = det(\7J, \7g). The nonlinear scalar evolution equation (2-14) is known to be well-posed for bounded measurable vorticity functions. More precisely, if the initial vorticity satisfies Wo E Loo(D) then w E Loo((O, (0) x D) and the Loo(D)-norm of w(t,') is preserved for all t > 0. This bound on vorticity provides adequate smoothness OIl the velocity field to guarantee existence and uniqueness of particle paths

dx dt

=

u(t, x),

x(O)

= Xo

ED,

from which it follows that there is a unique weak solution w(t,x) of equation (2-14) for any initial vorticity Wo E Loo(D). Even for smooth initial vorticity fields, however, the regularity of weak solutions to (2-14) quickly degenerates as time proceeds, owing to the rapid growth of the vorticity gradient. This growth results from the increasingly intricate spatial arrangements realized by

5. Turbulent fluids

39

the vorticity field as it is advected by the flow. This turbulent behavior is well-documented by numerous direct numerical simulations of high Reynolds number two-dimensional flows. Because of its highly complicated microscopic behavior, the vorticity field w(t, x) itself, therefore, does not provide a useful description of the long-time behavior of the fluid. For this reason, it is desirable to shift to a macroscopic description of the vorticity distribution that only partially encodes the rapidly increasing information content of the microscopic vorticity field. Such a description is afforded by the parametrized measure v = {VXLEO associated to the sequence of functions {w(tj,.)} when tj ---+ 00 for a weak solution of the Euler equation, w. Indeed, if Ilwollv"'(o) = r, then for all t > 0, Ilw(t, ')llv"'(o) :::; r. Therefore, we can find sequences tj ---+ 00 such that {w( tj, .)} generates a parametrized measure v = {vx } xEO with the support contained in the interval [-r, r]. This measure captures the limiting statistics of the sequence {w(tj,·)} in an infinitesimal neighborhood of each point in the spatial domain n. There may be many such parametrized measures depending on the particular sequence tj that is chosen. We wish, therefore, to select from the set of possible parametrized measure weak limits the one that is in some sense most likely to be realized. The first difficulty that is confronted in this program is that of determining an appropriate class of admissible parametrized measures. We must recognize that it is seemingly impossible to characterize completely the set of such families of measures that can be generated by sequences of vorticity functions corresponding to a solution of the Euler equations. This is due to the highly complex behavior exhibited by the vorticity field as it evolves, as alluded to above. Indeed, for all practical purposes, the only useful information that remains after a certain period of time is that the energy and entropy of the system are invariant under the dynamics. These quantities may be expressed as functionals on the vorticity field. The requisite formulas are, respectively,

E(w) =

~

In w'ljJ dx,

In f(w) dx,

Fj(w) =

where f can be any continuous function in [-r, r]. Notice that there is an infinite family of conserved entropy integrals. It is generally accepted that these are the only invariants of the dynamics, aside from those that may arise from special domain geometries. We shall assume that they exhaust the list of invariant functionals. The conservation of energy and entropy by the dynamics translates into corresponding constraints on the possible parametrized measure weak limits. Indeed, if EO and are the values of energy and entropy fixed by the initial vorticity Wo, then for Wj = w(tj,')

FJ

11

E(v) = lim E(wj) = A

J--->OO

2

0

w'ljJdx = E 0 , A

40

Chapter 2. Some Variational Problems

and where w is the weak limit of {Wj} (or of an appropriate subsequence) or the first moment of v, and ~ = Gw is the corresponding stream function. Notice that we have used the compactness of the Green's operator G. It should be noticed that the energy of v resides in the mean field w since E(v) = E(w) = E(wo) = EO; the fluctuations do not contribute to the energy. On the other hand, the microscopic fluctuations do contribute to the entropy integrals since in general it is not true that Ff(v) has the same value as Ff(w). We might say, therefore, that entropy is not conserved on a macroscopic scale, as part of it is lost to the infinitesimal-scale fluctuations of the vorticity. We have demonstrated that if the parametrized measure v is generated by a sequence of vorticity functions Wj arising from a solution of the Euler equations, then it must satisfy the above energy and entropy constraints. Insofar as this is the only tangible information available about the possible long-time weak limit parametrized measures, we shall take as our admissible class of measures the set A =

0

0

v = {vXLEn : supp (v x ) C [-r, r], E(v) = E ,Ff(v) = F f for all A

{

A

f} .

We now seek to determine those elements in A that are in some sense most probable, and therefore the most likely to be observed as long-time equilibrium states of the Euler system. This task is accomplished through the introduction of the Kullback entropy functional K7r(v) = -

r1

in [-r,r]

log

~vx

7l'o

dVx(Y) dx

if Vx is absolutely continuous with respect to 7l'o. Otherwise it is taken to be -00. Here 7l'o is a fixed probability measure on [-r, r] and 7l' = dx Q97l'o is a spatially homogeneous probability measure. The functional K is well known from information theory and statistical physics. As an integral in y it is a measure of the logarithm of the number of microscopic vorticity fields W corresponding to the macrostate v. The functional I = -K is a measure of the statistical distance from v to the homogeneous parametrized measure 7l'. Thus if v maximizes K over the admissible class A, then v minimize::; the di::;tance to 7l' and v is also most probable in the sense that it corresponds to the largest number of microstates w. It is clear that the choice of the reference measure 7l' is important. It has been argued that 7l'o should be chosen to be the probability measure (1/ IOI)7l'w, where 7l'w is the vorticity distribution function defined by

6. Bibliographical remarks

41

This distribution function is conserved by the Eulerian flow, because the entropy integrals are invariant. The measure 7r then represents the most mixed, or most random macrostate. It satisfies the entropy constraints, but not the constraint on the energy. With this choice of 7r, the most probable parametrized measure consistent with both of these constraints is determined as a solution of the maximum entropy principle

Kn:(v)

---t

max

subject to v E A.

While we have attempted to motivate the maximum entropy principle as an intutitively appealing procedure for selecting the most probable admissible parametrized measures, its rigorous justification rests upon methods from statistical mechanics and the theory of large deviations. These developments are beyond the scope of this text. We merely wish to point out that the set of solutions of the maximum entropy principle, A*, satisfies a natural concentration property, which roughly states that an overwhelming majority of the measures in the admissible class A concentrate about that subset of solutions. In particular, any parametrized measure that is generated by a sequence of vorticity fields corresponding to a solution of the Euler equations concentrates about

A*. 6. Bibliographical remarks References for Sections 2 and 3 are basic works on parametrized measures, the calculus of variations and optimization; these have already been mentioned in Chapter 1. Sections 4 and 5 are, however, more specific. An important subject from the point of view of applications not included in this chapter where weak convergence and homogenization play also a fundamental role is the theory of composites. A few references on this topic are [11], [146], [197], [235], [236],

[237].

The main sources for the optimal design problem of Section 4 are [49], [50], [51], [198]. Numerical experiments are recorded in [71], [72]. The optimal

relaxation as well as the general approach in terms of parametrized measure as it has been explained in this chapter can be found in [245]. [246] and [301] contain the basic results on H-convergence used in this problem. A more detailed discussion of the statistical approach in terms of parametrized measures of turbulence as well as justification for some of our remarks in Section 5 can be studied in [28], [48], [119], [165], [177], [178], [180], [234]'

[273], [311].

Chapter 3 The Calculus of Variations under Convexity Assumptions

1. Introduction

The central focus of the calculus of variations is the functional

J(u)

=

1 n

cp(x, u(x), Vu(x)) dx,

where the integrand cp explicitly depends upon the gradient variable Vu. n is assumed to be an open, regular, bounded domain of RN. The admissible functions u : n ---t R m belong to some reflexive Sobolev space and they may satisfy some other restriction like having the boundary values prescribed. The integrand cp : n x Rm x MmxN ---t R* is assumed to be a Caratheodory function. By this we simply mean that cp is measurable on the x variable and continuous with respect to u and Vu. We may eventually let cp take on the value +00 as indicated by R* = R u {+oo}. We devote the present chapter to proving results in the spirit of Theorem 1.1 for this type of functionals. The main difficulty is the weak lower semicontinuity property. We want to understand the conditions on cp that ensure this important property. This will take us to the quasiconvexity condition for 'P, so that gradient parametrized measures will also playa crucial role in the analysis that follows. Since the quasiconvexity condition, except for the scalar case, is hard to grasp we look for sufficient conditions for quasiconvexity. Polyconvexity is then introduced as the main source of quasiconvex functions that are not convex. Our analysis does not pretend in any way to be complete in this regard. Having in mind applications of existence theorems for polyconvex integrands, we discuss very briefly three-dimem;ional elasticity in Section 5. Finally we explore how the fact of being a minimizing sequence for some functional provides further information that can be used to derive weak and strong convergence results and representation formulas in terms of gradient parametrized measures. Remember that a Wl,P-parametrized measure is the parametrized measure associated to a bounded sequence of gradients in LP(n). P. Pedregal, Parametrized Measures and Variational Principles © Birkhäuser Verlag 1997

44

Chapter 3. The Calculus of Variations under Convexity Assumptions

The proofs in this chapter are based on the results stated in Chapter 1, Sections 4 and 5. The complete proofs of those are contained in Chapters 6, 7 and 8. Because the space W1,1(n) is not reflexive, the case p = 1 is very special. Even though some of the conclusions in this chapter are valid for p = 1, or may be restated in some way so that they become true, we consistently avoid this delicate case. We take 1 < p < 00 throughout this chapter unless explicitly stated otherwise. 2. Weak lower semicontinuity We start by giving the proof of a very general weak lower semicontinuity result for functionals I of the type described in the Introduction. We first consider the integrand cp depending on the gradient variable alone and move on to the case of full generality.

Let cp be a continuous function defined over matrices, bounded from below. Let {Uj} be a sequence of W1,P(n)-functions converging weakly in W1,p(n) to u. Let v = {vx}xEfl be the parametrized measure associated to {Y'Uj} (or possibly to a subsequence), so that

Theorem 3.1

Y'u(x) = (

Advx(A),

JM'mXN

Ifliminfj_Hx'!ncp(Y'Uj)dx <

00,

a.e. x E

n.

then

rcp(Y'u)dx ~ liminf JEr
JE

)--+00

for all measurable E c n, if and only if cp(Y'U(x))

~

(

cp(A) dvx(A),

JMmxN

a.e. x E

n.

(3-1)

Proof Recalling Theorem 6.11 and assuming Jensen's inequality (3-1) we have

liminf ( cp(Y'Uj) dx '2 {

)--+00 JE

(

JEJM'mXN

'2

L

cp(A) dVx(A) dx

cp(Y'u(x)) dx.

Conversely, if {cp(Y'Uj)} is a uniformly bounded sequence in L1(n), by Chacon's biting lemma there exists a nested sequence of measurable sets nk whose measures tend to 0, and such that

cp(Y'Uj) ~ VJ in Ll(n \ n k ), for all k, VJ(x) =

r

JM'mXN

'P(A) dVx(A).

45

2. Weak lower semicontinuity

By assumption

= /'

JE\r:lk


By Theorem 6.11,

r
k,

JE

M'nxN


•

The arbitrariness of E yields (3-1).

The proof of this theorem in the general case in which

Theorem 3.2 Let

V'U(x)

=

1

MmxN

Advx(A).

If {
r

1


forallEcSl,

a.e. x

E

Sl.

(3-2)

MTnXN

For the proof simply notice that by Proposition 6.13 and the comments after it, the parametrized measure associated to {( Uj, V'Uj)} is {Du(x) Q9 vx } xH! and follow the steps of the previous proof. The two theorems above are concerned with a particular sequence of functions and a particular energy density so that if inequality (3-2) is verified for a given parametrized measure and energy density
46

Chapter 3. The Calculus of Variations under Convexity Assumptions

in W 1,P(0) converging weakly, we would obtain the following "convexity" condition on the energy density t.p:

r

JM~XN

t.p(x, u, A) dv(A) 2 t.p (x, u,

r

lMmxN

A dV(A)) ,

(3-3)

for all (x, u) EO x Rm and for all v, a homogeneous W 1,P-parametrized measure. Notice that by the localization principle Theorem 8.4 each Vx can be regarded as a homogeneous W 1,P-parametrized measure by itself. This convexity condition (3-3) takes us one step further than quasiconvexity or W 1'P-quasiconvexity. Let us assume that t.p is bounded from below and depends upon the gradient variable alone. Take U E W 1 ,P(0) such that U - Uy E WJ'P(O) where uy(x) = Y x and Y is any fixed matrix. We can consider the average of 8Vu (x) , vu , which is a homogeneous W1,P-parametrized measure with underlying deformation Y according to Theorem 8.1. For v u , (3-3) reduces to

t.p(Y) :S

l~l

In

t.p(\i'u(x)) dx,

which is the usual W 1,P-quasiconvexity condition. In (3-3), however, we are demanding the inequality for "all" homogeneous W1,P-parametrized measures and not just for those coming from single functions. In so doing, we are requiring that concentration effects do not have any influence on the quasiconvexity inequality when sequences are considered. In order to better explain this issue, suppose we have a bounded sequence {Uj} C W 1 ,P(0) with Uj -Uy E WJ'P(O). By the quasiconvexity condition

t.p(Y):S

l~~~f l~l

In

t.p(\i'uj(x))dx.

On the other hand if v = {vx } xEn is the parametrized measure associated to {\i'Uj}, since we have affine boundary conditions, we can consider the averaged, homogeneous W 1,P-parametrized measure, v. If the sequence {t.p(\i'Uj)} does not develop concentrations, so that it is a weak convergent sequence in L1 (0), then (3-3) is a corollary of the usual W 1'P-quasiconvexity condition because in this case for a suitable subsequence

iM'rmXN t.p(A) dV(A) =

lim

l~l inr t.p(\i'Uj(x)) dx,

J ..... OO H

and (3-3) holds. If, on the other hand, {t.p(\i'Uj)} develops concentrations, then

r t.p(A) dV(A) < iMmxN but still (3-3) requires

lim

l~l inr t.p(\i'Uj (x)) dx,

J ..... OO H

2. Weak lower semicontinuity

Theorem 3.3

47

Consider the functional J(U)

=

10 'P(x,u(x), V'u(x))dx

(3-4)

defined on W1,P(fl) where 'P is non-negative and can eventually take on the value +00. J is weak lower semicontinuous in W1,P(fl) over arbitrary measurable sets if and only if 'P is "convex" in the sense just described. Proof i) Necessity. Let (xo, un) E fl x Rm be given and v, a homogeneous Wl,P-parametrized measure generated by {V'Uj}. Set Y = fwnxN Adv and u(x) ::= Y(x - xo) + un. By adding suitable constants to Uj we may assume without loss of generality that {Uj} converges weakly in W1,P(fl) and strongly in U(fl) to u. By Theorem 3.2,

r

JMmxN

'P(xo, un, A) dv(A) ;::: 'P(xo, un, Y).

ii) Sufficiency. Let {Uj} be any sequence converging weakly in W1,P(fl) and strongly in LP(fl) to U and let v = {vx } xEO be the parametrized measure associated to {V'Uj}. By the localization property (Theorem 8.4) we know that each individual Vx is a homogeneous Wl,P-parametrized measure, therefore by the "convexity" property (3-3) for U = u(x) and v = Vx

r

JMTnXN

'P(x, u(x), A) dVx(A) ;::: 'P(x, u(x), V'u(x)),

for a.e. x E fl. By Theorem 3.2, the weak lower semi continuity property holds over arbitrary measurable sets. • We describe below three situations in which this "convexity" condition can be translated into a more familiar convexity condition.

Theorem 3.4

Let'P: fl x Rm x MmxN ---> R be a finite, continuous energy density. The functional J(u) in (3-4) is lower semicontinuous with respect to weak convergence in W1,OO(fl) over arbitrary measurable sets if and only if 'P(xo,uo,') is quasiconvex for any Xo Efland Uo E Rm. Proof Let Y be any matrix and U E W1,Q()(fl) such that U - Uy E WJ,OO(fl). The averaged parametrized measure, v, corresponding to O-Vu(x) is a W1,oo_ parametrized measure. Therefore

I~I

in

'P(xo, un, V'u(x)) dx

= LmxN 'P(xo, uo, A) dv(A) ;:::'P(xo, uo, Y),

where Y =

LmxN Adv(A) =

and 'P(xo, uo,·) is quasiconvex.

I~I

in

V'u(x) dx,

48

Chapter 3. The Calculus of Variations under Convexity Assumptions

Conversely, let us assume that ¥?(xo, uo,') is quasiconvex. If v is any W 1,00_ parametrized measure, there exists a sequence of functions, {Uj}, in W 1,00(0) whose gradients generate v and Uj - Uy E W~,oo(O) for all j. In this case (because of the uniform boundedness of V Uj) {
{

iMmxN

¥?(xo, un, A) dv(A)

=

lim

)--00

I~I in(
•

by the quasiconvexity property. Theorem 3.5 Let

X

Rm x

MmxN

---7

R be a finite, continuous energy

o ::;
where h E L~AO x Rm) is continuous in U and 1 < p < 00. The functional J(u) in (3-4) is lower semicontinuous with respect to weak convergence in W 1 ,P(0) over arbitrary measurable sets if and only if
Proof By Theorem 3.3 we have to check again that under these assumptions quasiconvexity is equivalent to the "convexity" condition in terms of W 1 ,p_ parametrized measures. Assume that
l~r::~f 1~ll ¥?(xo, Un, VUj(x)) dx

= {

iMmxN

¥?(xo, un, A) dv(A).

The converse follows immediately from the same argument given in the previous theorem. • If we consider scalar problems where m = 1 or N = 1 the "convexity" condition is nothing but the classical Jensen's inequality. Indeed, in either of these two cases quasi convexity, and therefore rank-one convexity, reduces to convexity since all directions on M mxN are of rank-one. Thus the weak lower semicontinuity property is valid if and only if the integrand is convex in the usual

sense.

3. Existence theorems

49

Theorem 3.6 Let rp : 0 x R X M 1xN -+ R* (respectively rp 0 x Rm x Mmxl -+ R*, 0 C R) be a Caratbeodory function. The functional I(u) where the competing functions are scalar valued u : n c RN -+ R (respectively u: nCR -+ Rm), is lower semicontinuous with respect to weak convergence in W1,P(O), 1 ::; p::; 00, over arbitrary measurable sets ifand only ifrp(xo, uo,') is convex for a.e. Xo E 0 and every Uo E R (respectively Uo E Rm). If we do not assume any growth on rp in the vector case one would desire that the "convexity" condition be equivalent to W1'P-quasiconvexity, or at least that W1,P-quasiconvexity is the condition necessary and sufficient in order to have weak lower semicontinuity over regular, open subsets of O. Either of these two questions is still open.

3. Existence theorems Bringing together these weak lower semicontinuity results and the abstract setting for the direct method discussed in Chapter 1, one gets immediately the following existence theorems. The first one is concerned with the scalar case when 0 C R or the competing functions u are scalar-valued. The other two correspond to the fully vector case in two situations: under growth restrictions on the energy density and without these conditions. In all three theorems, rp : 0 x Rm x MmxN -+ R* will stand as usual for a non-negative energy density. Either of the cases m = 1 or N = 1 correspond to the scalar case. The functional we are considering is again

I(u) =

In

rp(x, u(x), V'u(x)) dx,

and the variational principle for which we would like to prove existence theorerns is

(P)

for some prescribed Uo E W1,P(O) such that I(uo) < set of admissible functions is A

=

00.

In this situation the

{u E W1,P(O) : u- Uo E W5'P(O)} which is

weakly closed in W1,P(O). The condition on the reflexivity of the underlying Banach space prevents us from taking p = 1, so that 1 < p < 00 as usual. The case p = 00 is also discarded because bounds in the W1,oo-norm for a minimizing sequence cannot be obtained from coercivity bounds on the energy density. Theorem 3.7 Let m = 1 or N on the gradient variable and

= 1. Let rp be a Caratheodory function, convex

max(O, c IAI P

-

1) ::; rp(x, u, A),

for a.e. x E 0 and all u E R rn , where c principle (P) admits at least a minimizer.

>

0 is a constant. The variational

50

Chapter 3. The Calculus of Variations under Convexity Assumptions

The coercivity property for the functional is obtained from the lower bound on

1 and N > 1. If

-

1) ::;
+ IAI P ),

where h E L~c(n x Rm) and c > 0, the variational principle (P) admits at least a minimizer.

If we drop the upper bound on
and

Let m > 1 and N > 1. If

max(O,cIAI P -1) ::;
1

MtnXN


MntXN

Adll(A)),

for all (x, u) E nxRm and for all II, homogeneous W1,P-parametrized measures, the variational principle (P) admits at least a minimizer. At this point, it should be clear how important convexity conditions are in order to have weak lower semicontinuity in different contexts. In the vector case, we have sufficiently emphasized that the quasiconvexity condition is the appropriate notion. Unfortunately, in practice, it is almost impossible to decide whether an explicit function is quasiconvex and almost impossible to compute the quasiconvexification even in the simplest, nontrivial cases. The difficulty is related to the essentially nonlocal nature of the definition of quasiconvexity and our inability to understand how gradients are built. Since the pioneering work of Morrey, there have been several attempts to further understand the notion of quasiconvexity, and in particular to decide whether quasiconvexity is equivalent to rank-one convexity. As we now know, rank-one convexity is weaker than quasiconvexity. We will show this in Chapter 9. There are some situations, though, where by restricting further the form of the function one can show that rank-one convexity is indeed equivalent to quasiconvexity. The most important example is when

4. Polyconvexity

51

4. Polyconvexity To explain this new convexity, one needs to identify the weak continuous functionals: the continuous functions '{J defined on matrices, such that both '{J and -'{J are quasiconvex. In this case we would have that if Uj ...:. U in W1,OO(n), then

Under suitable upper estimates on '{J, we could have the same weak continuity result for weak convergence in W1,P(O). A first idea is that since quasiconvexity implies rank-one convexity, '{J and -'{J will have to be rank-one convex, i.e. the function '{J(A + tB) must be affine in t provided B has rank one. The functions that enjoy this property are called quasi affine functions and they are identified in the following theorem which we do not prove here. Theorem 3.10 tions.

The minors of a gradient matrix are the only quasiafHne func-

In the case of 2 x 2 matrices, there are five minors: the four entries (the 1 x 1 minors) and the determinant. For 3 x 3 matrices, we have nineteen minors: the nine entries, nine 2 x 2 minors and the unique 3 x 3 minor, the determinant. And so on. The fact that these minors are quasi affine is easy to check. As a matter of fact, it is not hard to convince oneself that for one of these minors, 1fJ, the function of t '{J(A + tB) is a polynomial of degree rank(B). That these are the only quasi affine functions requires more work and a little bit of algebraic manipulation. See the last section for more references. We know where to look for the functions which give rise to weak continuous functionals. By an abuse of language, we refer to those functions as weak continuous functions. In order to be more precise, we need to introduce some notation. Let A stand for any matrix in MffiXN and A' for any square submatrix of A. adj(A') is the matrix of cofadors of A' so that A' adj(A') = det(A') 1, where 1 is the identity matrix in the appropriate dimension. If A' is nonsingular, then adj(A') = det(A')(A,)-l. Let

T

= max{m,N} and p

Theorem 3.11

Ifuj -'

U

> T. in W1,P(O) then det(V'uj)' -' det(V'u)' in LP/r(o).

52

Chapter 3. The Calculus of Variations under Convexity Assumptions

Proof We divide the proof in several steps. Step 1. Let v E W1,P(D). We claim that div (adj(V'v)') = 0 in the sense of distributions. Assume first that v is actually smooth. Based on the equality of the mixed partial derivatives, we find indeed that div (adj(V'v)') = O. For a general v E W1,P(D), take a sequence of smooth functions, {Vj}, converging strongly to v in W1,P(D). adj(V'vj)' converges strongly to adj(V'v)' in LP/r(D) (using Holder's inequality) because the terms in adj(V'vj)' are products of at most r - 1 factors. For a smooth test function 'IjJ

In

adj(V'vj)'V''ljJdx = 0,

for all j. By the strong convergence just pointed out

In

adj(V'v)'V''ljJdx

= 0,

and this is our claim. Step 2. As a consequence of Step 1, we obtain that div'(u adj(V'u)')

= (V'u)'

adj(V'u)'

= det(V'u)'

as distributions where div' means divergence with respect to the variables involved in the submatrix A' that has been determined previously. By induction, let us assume that adj(V'uj)' ~ adj(V'u)' in £P/r(D). If 'IjJ is a smooth test function, then

In det(V'uj)''ljJdx In =

Uj adj(V'uj)'V''ljJdx.

(3-5)

But {Uj} converges strongly to U in Loo(D) by the Compactness Theorem for Sobolev functions (p> N). Hence the limit in (3-5) is

In

U adj(V'u)'V''ljJdx =

10 det(V'u)''ljJdx,

and det(V'uj)' converges weakly in the sense of distributions to det(V'u)'. Step 3. Conclusion. We also have a uniform bound on det(V'uj)' in LP/r(D) because terms in det(V'uj)' have less than r factors. Since plr > 1, at least for a subsequence (not relabeled) det(V'uj)' converges weakly in £P/r(D). By Step 2 and the uniqueness of the limit we conclude that in fact det(V'uj)' converges weakly in LP/r(D) to det(V'u)'. • This result holds for r = min {m, N} as well. The proof is the same. It only requires a more careful analysis of exponents. Let M(A) represent the vector of all possible minors of any dimension of A considered in some order. A continuous function 'P : MmxN ---> R* is called polyconvex if it can be rewriten as g(M(A)) where g is a convex function of all its arguments (convex in the usual sense). The most important property of polyconvex functions is that they are quasiconvex.

4. Polyconvexity

53

Proposition 3.12 Let r be a polyconvex function. For p ~ r, r satisfies Jensen's inequality (3-3) for any homogeneous Wl,P-parametrized measure.

Proof The proof is simple. Assume that we have a uniformly bounded sequence in W1,P(O), {Uj}, generating a homogeneous parametrized measure v with first moment Y: Y = A dv(A). Without loss of generality we may well assume that {1V'uj is equiintegrable according to Lemma 8.15. Since {Uj} converges weakly to U y, affine, by the previously established weak convergence for p > r,

n

J

This is also true for p = r by the assumed equiintegrability and the fact that minors of any order arc bounded above by the power corresponding to its order, IM(A)I S; C(l

+ IAn·

Because of this weak convergence the representation in terms of v is valid

Hence

1

M(A) dv(A) = M(Y) = M

M~xN

(1

A dV(A)) .

M~xN

Since 9 is convex, by Jensen's inequality,

L"'XN r(A) dv(A) = LmxN g(M(A)) dv(A) ~g (LmxN M(A) dV(A)) =g(M(Y)) =r(Y)'

•

As a consequence, any polyconvex function is quasiconvex. We can now write down many non-trivial examples of quasiconvex functions. For example, in the case m = N, any convex function of the determinant is quasiconvex (notice that the determinant itself is not convex). One particularly important example is the jacobian: r(A) = Idet AI. Because of the upper bound on the determinant

we conclude that the jacobian is W1,P-quasiconvex for p ~ N. If we are willing to accept also dependence of r on x and u, polyconvexity is defined in the same way for a.e. x E 0 and all U E Rm. We have the following existence theorem for polyconvex integrands which is a corollary of Theorem 3.9.

54

Chapter 3. The Calculus of Variations under Convexity Assumptions

Theorem 3.13 If r.p : n x Rm x MmxN and for p ~ r (r = max {m, N}) c IAI P

-

---+

R* is nonnegative and polyconvex,

1 ~ r.p(x, u, A),

c> 0,

for all A E MmxN, a.e. x E n and all u E Rm, the variational principle (P) with integrand r.p admits at least a minimizer. More precise statements about polyconvexity and existence theorems can be found in the references (see Section 7). In the last few years a fairly large amount of work has been done trying to relate and understand all these different notions of convexity. In particular, counterexamples have been produced to show that quasiconvexity is strictly stronger than polyconvexity. We refer the reader to the bibliography.

5. A brief account of nonlinear elasticity This section presumes to be only a very short and basic review of the mathematical theory of nonlinear elasticity. The aim is to emphasize the importance of variational principles for the vector case, and the crucial role that polyconvexity plays. For the sake of brevity, we will not make precise statements. There are materials in nature whose equilibrium configurations in various enviroments can be understood through an energy minimization principle. The material seeks the minimum energy available to it under the prescribed conditions. In this sense we identify minimizers of the energy functional with equilibrium states. The possible deformations that a material may undergo are described mathematically by means of a vector function u : n ---+ R3 where n c R3 is the reference configuration with respect to which we consider all deformations. The gradient Vu is referred to as the deformation gradient and intuitively represents the local deformation or strain around each point x in the reference configuration. For the type of materials we are interested in, we assume the existence of a continuous stored energy density r.p defined on 3 x 3 matrices so that the free energy associated to a particular deformation is measured by the integral

10 r.p(Vu(x)) dx.

From the physical point of view, the energy density r.p must comply with several restrictions. For instance, it should be material frame-indifferent. We must require

r.p(F) = r.p(QF),

for all proper rotations Q in space (by proper we mean positive determinant rotations). Moreover, r.p must also satisfy the condition

r.p(F)

---+

+00 if

det F

---+

0,

r.p(F) = +00 if det F

~

0,

5. A brief account of nonlinear ela..'lticity

55

to reflect the fact that infinite energy is associated with "extreme" deformations trying to collapse some volume into a plane or a line, although this condition is often relaxed. Further restrictions can be imposed depending on specific properties of the material under consideration. These constraints have deep implications concerning the structure of cp. One of the most common situations consists in determining the equilibrium configurations of the material under prescribed boundary values. This is accomplished by determining the boundary values on an that competing deformations should have. In this framework, equilibrium configurations will correspond to minimizers of the variational principle

1

cp(Vu(x)) d:r:,

U

E

w1,p(n), u -

Uo

E

wJ,p(n).

12

We are faced with a variational problem of the type we have been discussing so far. The existence of such equilibrium states is closely connected to the "convexity" properties of the energy density cp. One striking consequence of the behavior of the energy dem;ity for minimizing deformations is that cp cannot be convex in the usual sense. This reasonable assumption rules out the possibility of having convex energy densities. The axiom of frame-indifference also has serious implications concerning the eigenvalues of the Cauchy stress tensor VuTVu. A crucial ohservation is that these difficulties are not present when considering polyconvex stored energy functions cp. An important class of polyconvex functions that appear as energy densities in nonlinear elasticity is

cp(F) =

L t

(Ii

tr(FT F),,';2

+

L tr(adjFT F/'j/2 + g(det F), "

j=!

i=1

where tr stands for the traee of a matrix, s, t are positive integers, (Ii > 0, Cti 2> 1, (1j 2> 1, and 9 is a convex function. This function satisfies a coerciveness inequality as well,

cp(F) 2>

Ct

(IIFII P

+ IladjFll q ) + g(det F),

where Ct > 0, p = max Ct; and q = lIlax (1j. A material whose energy density is of the above type and satisfies the additional property lim A---7o g(>..) = +00 is called an Ogden material. Particular examples are: l. Neo-Hookean materials:

cp(F) =

(I

IIFI12

+ g(det F),

a. > 0.

2. Mooney-Rivlin materials:

cp(F) = a IIFI12

+ b IIadjFI1 2 + g(det F),

(I>

0, b > O.

56

Chapter 3. The Calculus of Variations under Convexity Assumptions

For materials that admit this sort of stored energy density, the existence of equilibrium configurations can be easily established in the framework of the direct method of the calculus of variations. There are, however, some examples for which the energy density is not polyconvex. One such example is the St. Venant-Kirchhoff materials:


= a(trE)2 + b trE2,

1 + 2E

= FT F.

6. Weak and strong convergence of minimizing sequences Convergence of energies together with weak convergence enable us to obtain some useful weak convergence results in L 1 (0). This fact has important consequences for representation formulas in terms of parametrized measures for minimizing sequences. The issue we address is how the fact that a particular sequence is minimizing for some functional may help in deriving weak convergence results. Theorem 3.14

Let

Suppose that

1 n

Uj ...:':.. U in W 1,P(0),

cp(\lu) dx

=

lim

)-00

1 n

cp(\lUj) dx.

There exists a subsequence (not relabeled) such that

Proof Assume, by contradiction, that { 0 and every 8 > 0, there exists Ao c 0 and an integer jo such that IAol < 8 and

E

j

CP(\lUj6) dx >

E.

A6

Since
l


> 0 such that < E.

if lEI < 80 , then

(3-6)

Let us choose in particular 8k = 2- k80 • There exists a sequence A k, IAkl < 8k, and jk such that

6. Weak and strong convergence of minimizing sequences

Let E = Uk A k , so that

Letting jk ---+ energies that

00,

lEI:; 60

57

and (3 6) holds. Thus

we have by weak lower semicontinuity and the convergence of

(::; /' i.p(Vu)dx -

.fn

=

r i.p(Vu) dx

/'

.fn\E

i.p(Vu)dx

.fE

< f,

•

a contradiction.

The condition on the convergence of energies is automatic for minimizing sequences. The weak convergence of Theorem 3.14 together with the coercivity of i.p can be used to obtain representation of weak limits through parametrized measures associated with minimizing sequences. Let XP be the space of functions defined on matrices with growth of order less than p:

XP = {i.p : M mxN Corollary 3.15

---+

R, continuous: 1i.p(A)1 :; C (1

+ IAI P )}.

Let i.p be a quasiconvex function such that

max(O, c IAI P

-

1) :; i.p(A) :; C(IAI P

+ 1),

1:; p <

00,

0< c:; C.

Consider the variational principle

for Uo E W l.p (n). If { Uj} is minimizing, Uj ~ U (so that U is a minimizer) and v = {vX}rEn is the parametrized measure corresponding to {VUj}, then

and for any 1/) E XP where, as usual,

1j}(x)

=

1

MmxN

?jJ(A) dVx(A).

58

Chapter 3. The Calculus of Variations under Convexity Assumptions

The proof is a direct consequence of the previous theorem and is left to the reader. We would also like to give some conditions that would improve weak to strong convergence. These involve the notion of strict quasiconvexity. We say that a continuous function defined over the space of matrices MmxN is uniformly strict Wl,P-quasiconvex if there is a I > such that

+I

cp(Y)

L

l\7u - YIP dx S

L

°

cp(\7u) dx,

for any U E W1,P(O) with U - Uy E W~'P(O). This condition is connected to the study of partial regularity for minimizers of vector variational principles. A somewhat less restrictive notion, which we simply call strict W1,p_ quasiconvexity, requires that for any homogeneous W1,P-parametrized measure, v, with underlying deformation Y, the strict inequality cp(Y) <

LmxN cp(A) dv(A)

holds unless v = by. The fact that we can understand the gradient of any W1,P(O) function with affine boundary values Y as a homogeneous W1,p_ parametrized measure through the averaging procedure tells us that uniformly strict W1'P-quasiconvexity truly implies strict W1,P-quasiconvexity. Theorem 3.16 Ifuj ~ vex function satisfying

U

in W1,P(O), cp i8 a continuous, 8trict W1'P-quasicon-

max(O, c IAI P - 1) S cp(A) S C(1 and

then Uj

Inrcp(\7u)dx= -+

lim J--+OO

+ IAI P),

0< c S C,

Inrcp(\7uj)dx,

n strongly in W1,P(O).

Proof Using the weak lower semicontinuity for any measurable E complement 0 \ E we obtain

r cp(\7n)dx S liminf JEr cp(\7uj)dx,

JE

r

c 0 and its

In\E

J--+OO

cp(\7n) dx slim inf )--+00

r

In\E

cp(\7 Uj ) dx.

Coupling these two inequalities, by hypothesis we get, in fact, equality for the whole sequence. Therefore we should have

r

JE

cp(\7n)dx

= lim

r

)--+OOJE

cp(\7uj)dx,

59

7. Bibliographical remarks for any E

if v

c O.

= {vx }xEfl

On the other hand, by Theorem 3.5,

is the parametrized measure associated to {V'Uj}. For any E

But in fact, by Theorem 6.11, equality must hold

and in particular

Hence

cp(V'U(x)) =

r

JMmxN

cp(A) dvx(A),

a.e. x E O.

Since by the localization property, Theorem 8.4, each individual vx is a homogeneous W 1 ,P-parametrized measure, the strict W 1 'P-quasiconvexity of cp, let us conclude that Vx = D'\7u(x) for a.e. x E O. By Theorem 3.14, {cp(V'Uj)} (or some appropriate subsequence) is weakly convergent in £1(0). The coercivity assumed on cp enables us to obtain the equiintegrability on the powers {IV'uj IP }. Proposition 6.12 helps us in finishing the proof. • An immediate consequence of this theorem is that minimizing sequences for strict W1,P-quasiconvex integrands with the proper bounds always converge strongly in W1,P(0) to minimizers.

7. Bibliographical remarks The weak lower semicontinuity property is bound to convexity properties. Almost all of the references related to quasiconvexity treat the problem of weak lower semicontinuity under several settings. [3], [4], [29], [33], [37], [38], [97],

[106], [151]' [166], [193], [201]' [216], [220], [221], [225]' [233], [238], [240], [269], [272], [289] is a partial list of some basic papers. The books referred to in Chap-

ter 1 that take a point of view based on weak convergence are also references to keep in mind for convexity, quasiconvexity and lower semicontinuity. More general treatments of weak lower semicontinuity in the context of BV functions or

60

Chapter 3. The Calculus of Variations under Convexity Assumptions

with respect to measures have considered systematically by the Italian and the French schools: [12], [13], [14], [15], [16], [18], [52], [53], [54], [106], [141], to mentionjust a few. See also [143]. The approach taken here based on parametrized measures was mainly inspired by [186], [191], [260]. A similar point of view is taken in [40]. Polyconvexity was studied for the first time systematically in [29]. The fundamental observation that some of the difficulties in nonlinear elasticity are solved by considering polyconvex stored energy densities was also made in [29]. Existence theorems are usually obtained as indicated as soon as weak lower semicontinuity results are available. What is not so elementary is achieving existence results in spite of nonconvexity. For this problem there is abundant bibliography: [20], [21], [22], [23], [25], [26], [65], [66], [67], [68], [69], [99], [102], [117], [156], [217], [218]' [223], [226], [227], [228], [229], [266], [267], [292]' [300]. Quasiaffine functions have been also studied extensively because of their remarkable properties. Some works in this direction are: [33], [167], [168], [240], [241], [242], [269]-[272]. A remarkable weak lower semicontinuity counterexample for the jacobian is contained in [215]. Concerning continuum mechanics and elasticity, there are some good introductory textbooks like [78] and [163]. The material in Section 5 has been taken from [78]. Some other references in this direction are [17], [29], [36]. The results in the last section have been taken from [190]. The problem of the regularity of minimizers under conditions of strict convexity has been studied in [5], [30], [133], [134], [159], [222].

Chapter 4 N onconvexity and Relaxation

1.

Introduction

We have seen in the previous chapter that the weak lower semicontinuity property is crucial in order to employ the direct method of the calculus of variations to find minimi~ers of variational principles. This property is inherited by functionals whose integrands enjoy the appropriate convexity. Nonetheless, for an ever increasing number of interesting problems these convexity properties fail. In some cases, specific techniques may provide solutions to problems. In some others, this lack of convexity is a precursor of nonsolvability, at least in a classical sense. In the latter, highly oscillatory phenomena are usually involved. Parametrized measures were originally introduced by Young to account for oscillations in nonconvex optimal control problems where one could not reasonably expect classical i:iOllltions. To explain the nature of oscillatory minimizing sequences, let us look at a very simple, one-dimensional example eHsentially due to Bolza. The functional is

I(l1.) =

t

Jo

[ip(n'(x))

+ 11.(X)2]

d:J:,

where ip(t) = (t 2 ~ 1)2. The boundary conditions we take for admissibility are 11.(0) = n(l) = O. We notice that there is some kind of conflict between the two contributions to I. On the one hand the term on the derivative would favor slopes +1 or ~ 1, while the quadratic term would be minimized when 11. = 0 which does not have either of the two slopes. This incompatibility is responsible for the behavior of minimizing sequences and Ultimately for the lack of minimi~ers for I. This idea can be made precise in the following way. Step 1. The infimum of I under the given boundary conditions is O. For this, it is enough to take the saw-tooth functions

11.dx) =

~~ 2k

Ix

~

(kx) k

~ ~I. 2k

(Remember that (a) stands for the integer part of a E R). It is an easy exercise to check that this formula gives functions with slopes + 1 and ~ 1 and fulfills the P. Pedregal, Parametrized Measures and Variational Principles © Birkhäuser Verlag 1997

62

Chapter 4. Nonconvexity and Relaxation

°

boundary conditions. Moreover, since {ud converges strongly in the Loo-norm to we get I(Uk) --+ 0. Indeed {Uk} converges weakly in W 1,OO(0, 1) to 0. Step 2. I does not admit minimizers because if there exists a function U such that I(u) = 0, on the one hand u == but on the other hand the slopes that u may use should be restricted to ±1 and this is obviously impossible. In particular, even though the minimizing sequence {ud converges weakly to 0, the function is not a minimizer so that I cannot be weakly lower semicontinuous. Observe that rp is not convex. The behavior of any minimizing sequence for this problem is essentially the one described by {ud. The analysis of I may now proceed in two directions, although they are not, in fact, different directions. First, we can "convexify" the integrand rp and consider the regular variational problem associated with the convexification of

°

°

rp,

I(u)

=

10

1

[rp**(u'(x))

+ u(x)2]

dx,

where rp** is the convexification of rp. This new problem is usually referred to as the associated relaxed problem. The connection between the relaxed problem and the original one is established through the fact that the two infima, the one for I and the one for I, are equal (under suitable technical assumptions). In this specific case rp**(t) = {rp(t), It I :::: 1, 0, It I :s; 1, so that the two infima are easily seen to vanish. Notice that the function that identically vanishes in [0,1] is a minimizer for I, because I is weak lower semicontinuous, but it is not for I. The relaxed problem does not provide much information on the behavior of minimizing sequences. If one desires insight on that matter, the original problem must be embedded in a generalized variational principle where parametrized measures are allowed to compete in the minimization process. The functional becomes

[fa rp(A)dvx(A) +u(X)2] dx, v = {vxLESl' 10 fa A dVy(A) dy = 0,

J(V)

=

10

1

1

where

Notice that

J(8 Uf(X))

=

I(u),

for any admissible u. In this one-dimensional example there are no more restrictions on the competing v. In the vector case and in more general situations,

63

2. Relaxation theorem

understanding these constraints is the most important and most delicate question concerning the analysis of 1. Under suitable hypotheses, i admits always a parametrized measure solution and all admissible generating sequences are minimizing sequencec; for the original problem. In this sense we say that the behavior of minimizing sequencec; is encoded in the parametrized meac;ure solution for i. The oscillatory behavior we are attempting to examine is present in many interesting situations in models of continuum mechanics, problems of optimal design, optimal control and even turbulence phenomena and large time behavior for complicated systems. Some of these have already been examined in Chapter 2. They have served to motivate our way of understanding weak convergence through parametrized measures. In the next chapter some other fundamental situations are analyzed in greater detail. This chapter is devoted to the analysis of nonconvexity for the standard problem of the calculus of variations. A basic relaxation theorem is proved in Section 2 based on the program set forth in Chapter 1. An analysis of variational principles where we let parametrized measures compete follows. The main underlying ideas might also be applied to other problems that may not exactly fit in this category, as has been pointed out in the previous paragraph.

2. Relaxation theorem A typical relaxation theorem establishes, under some technical assumptions, that the infimum of any functional does not change when we replace the integrand by its ·'convexification". Theorem 4.1 such that

Let 'P :

n

x R= x MmxN

--->

R be a Caratheodory function,

where 0 < c, p > 1 and h is a locally bounded function. For any given W1,p(n), the two infima

inf

{L

'P(x, u(x), V'u(.1;)) dx : u - Uo E

w~,p(n)} ,

inf { ( Qcp(:c,u(x), V'u(x)) dx: u - Uo E

.In

Uo E

w~,p(n)},

are equal. The basic fact we need to prove this relaxation theorem is contained in the following lemma which by itself is a homogeneous version of the theorem.

64

Chapter 4. Nonconvexity and Relaxation

Lemma 4.2

Let'lj; : MmxN

----*

R bc continuous such that

c(IAI P -1) :::; 'lj;(A) :::; C(l

+

IAn,

p> 1,0

< c < C.

For any Y E M mxN there exists a homogeneous W1,P-parametrized measure, Vy, such that

=

Y

LmxN Advy(A),

Q'lj;(Y)

=

LmxN 'lj;(A) dvy(A).

l'vIoreover

Proof of lemma. Consider the following variational principle Q'lj;(Y)

= inf {

L

I~I

'lj;C'ilu) dx : U E W 1,p(n), U - Uy E

w~,p(n)} ,

and let {Uj} be a minimizing sequence. Since we have affine boundary conditions, by the average process Theorem 8.1, we may assume that the W 1 ,p_ parametrized measure associated to {'il Uj} is homogeneous, Vy, so that Uj ~ Uy in W 1 ,p(n). Then Y =

1

MmxN

Advy(A).

By Lemma 8.10 and Theorem 8.13, Q'lj; is quasiconvex, so that Q'lj;(Y) = inf

{I~I

L

Q'ljJ(V'u) dx : U E W 1,p(n), U - Uy E

w~,p(n)}.

(4-1)

Since for all j

we conclude that {Uj} is also minimizing in (4-1). By Lemma 8.12 Q'lj; inherits the same coercivity than 'lj; because the lower bound for 'lj; is a convex function and hence we have exactly the same lower bound for Q'lj;. Theorem 3.14 enables us to affirm that Q'lj;('ilUj) ~ Q'lj;(Y) in L 1 (D). By the coercivity {1'ilujIP} (or some subsequence) also converges weakly in L1(n). By the upper bound on 'lj;, the same is true for {1f;('ilUj)} and hence we have the representation

1121 Q'lj;(Y) = lim )->00

r 'lj;(A) dvy(A), illr'lj;('ilUj) dx = 1121 iMmxN

as desired. The fact that I·I P is integrable with respect to Vy is an immediate consequence of the bounds assumed on 'lj;. •

3. Parametrized measures solutions of variational principles

65

Proof of Theorem 4.1. Let m and Qm denote the two infima, respectively. Trivially Qm :::; m. In order to show equality, let U be any admissible function in W 1 ,P(0) so that U-Uo E WJ'P(O). By the bounds assumed on r.p and Lemma 4.2 we can find for a.e. x E 0, a homogeneous W 1,P-parametrized measure, v X , such that

r AdvX(A), Qr.p(x, u(x), V'u(x)) = r r.p(x, u(x), A) dvX(A). lMrnxN V'U(x) =

iwnxN

Consider the family of probability measures v = {vXLEn' We would like to show that 1I is a W1,P-parametrized measure. According to Theorem 8.16 we have to check three conditions. These hold essentially by construction. First, the fact that Jensen's inequality holds for quasiconvex functions in £P is true because each V X has been chosen to be a homogeneous Wl,P-parametrized measure. The compatibility condition that the first moment should be a gradient is also automatic. Finally the coercivity condition assumed on r.p yields the finiteness of the integral of the pth power against v. Thus, there exists a sequence offunctions in W 1 ,P(0), {Uj}, whose parametrized measure is precisely v = {VX}XEO and {1V'ujIP} is weakly convergent in £1(0) (Lemma 8.15). Once we have this weak convergence, we can assume that each Uj is admissible by Lemma 8.3. In this case lim

rr.p(x,Uj(x),V'uj(x))dx= ioriwnxN r r.p(x,u(x),A)dvX(A)dx = rQr.p(x, u(x), V'u(x)) dx. in

)-->00 in

The arbitrariness of U yields the result.

•

3. Parametrized measures solutions of variational principles We have already talked about parametrized measures solutions of variational principles in some of the examples in Chapter 2. We would like to examine from this point of view the standard problem of the calculus of variations under failure of the quasiconvexity condition for the integrand. Important applications will be discussed in Chapter 5. In many different models of mathematical physics we need to consider variational principles where the integrand r.p of the energy functional

J(U)

=

In

r.p(x, u(x), V'u(x)) dx,

66

Chapter 4. Nonconvexity and Relaxation

is not quasiconvex on the gradient variable. Uo is assumed to be some fixed function in W1,P(O). As pointed out, the typical behavior of minimizing sequences for these functionals is highly oscillatory: while the oscillations take place in regions of increasing fineness they remain of finite, nonvanishing amplitude. In these circumstances we talk about parametrized measures solutions. The assumptions for tp are the usual bounds

c(IAI P - 1) :::; tp(x, A, A) :::; C(1

+ IAI P + IAI P),

0 < c :::; C.

We would like to allow Wl,P-parametrized measures to compete in the energy minimization process. In order to do this, we define the energy of such a parametrized measure by

i(JL)

=

r1

ill

MmxN

tp(x, u(x), A) dJLx(A) dx,

where JL = {JLxLEI ll is a W1,P-parametrized measure generated by a sequence of gradients in W ,P(O), subject to the compatibility conditions

\7U(x) = ~iving

I.

1

MmxN

AdJLx(A),

the relationship between u and JL. We say that such a JL is admissible for

Note that we can always take JLx = 8vll (x) for some admissible u and in this case i(JL) = I( u). I admits a minimizing sequence {ud such that {1\7ukI P } is weakly convergent in Ll(O).

Lemma 4.3

Proof. Let {vd be any minimizing sequence for I. By the bounds assumed on tp, it is a bounded sequence in W1,P(O). Let v = {VXLEll denote the Wl,P-parametrized measure associated to the sequence of gradients {\7vd. By Lemma 8.15, v can also be generated by some other sequence of gradients {\7wk} such that {1\7wkjP} is weakly convergent in Ll(O). In particular, both sequences have the same weak limit in W1,P(O), u, and Wk -+ U strong in LP(O). By Lemma 8.3 we can find {ud admissible for I and still have the equiintegrability of {I \7 Uk IP }. Since {vd is minimizing

By Theorem 6.11 8trict inequality in the fir8t two terms is impossible, so that {Uk} is also minimizing. •

3. Parametrized measures solutions of variational principles

67

With this lemma we can now prove the following theorem.

Theorem 4.4 infI (u) = inf j (Ji) = inn (u), where I (u) is the energy fUIlctional whose energy density is the quasiconvexification of'P with respect to the gradient variable.

Prool Let m, in and m denote the three infima, respectively. In the previous section we have already shown that m = m. By the observation made prior to_ Lemma 4.3 we conclude that ih ::; m. To show equality, let It be admissible for 1. By Lemmas 8.15 and 8.3, we can find a sequence of W1,P(0)-functions, {Uj}, such that uJ - Uo E W~'P(O), {IVuj IP } is weakly convergent in Ll (0) and the parametrized measure associated to {VUj} is Ji. Thus lim l(uj) J--+OO

= lim /" 'P(x,11.j(x), V11.j(x))dx J-----c>x.ln

=1 /"

n .JMmXN

'P(x,u(.r),A) dltx(A) dx

=i(Ji) , where

VU(x) = /"

JMTnXN

Adltx(A).

•

This clearly implies that m = rh.

The advantage of dealing with 1 is that it admits minimizers within the class of Wl,P-parametrized measures only under the usual bounds on 'P. No convexity condition is needed or assumed.

Corollary 4.5

There cxists a v admissible for 1 such that

i(v) = int" i(p.).

Prvvl Take a minimizing sequence for 1, {Uj}, and let v be the parametrized measure associated to {VUj}. By Lemma 4.3 we may assume without loss of generality that {IVujjP} is weakly convergent in U(O), so that TTL

=

m = lim 1(11.j) = i(v). ]--+oc·

In this way we have a limit energy density for 1 "ip(.r)

~

/"

JMInXN

'P(x, u(x), A) dVI(A),

•

68

Chapter 4. Nonconvexity and Relaxation

where v = {v x } xEn is a minimizer for quantity'ljJ : M mxN ----+ R such that

i. Moreover for any continuous, nonlinear

we have a representation in terms of v

Given a non-convex functional I, we now have two ways to obtain a wellbehaved functional associated with it, I and J. A natural question is how minimizers for both functionals are related. Corollary 4.6

Let v be a minimizer for

V'u(x) = for

U

1

1. If

Advx(A),

MmxN

E W1,P(0), then u is a minimizer for

Qtp(x, u(x), V'u(x)) =

1

MmxN

a.e. x E 0,

(4-2)

I and

tp(x, u(x), A) dvx(A),

a.e. x E O.

(4-3)

Conversely, if u is a minimizer for I and v = {v x LEn is an admissible W1,p_ parametrized measure such that (4-2) and (4-3) hold, then v is a minimizer for 1. The proof is elementary. Simply notice that u is admissible, and by Jensen's inequality we can write the following chain of inequalities m ::; I(u)

=

: ; Inr r

::; Inr 1

L

JM=XN

Qtp(x, u(x), V'u(x)) dx Qtp(x,u(x),A) dVx(A) dx

M=xN

= J(v)

tp(x, u(x), A) dVx(A) dx

= m= m.

Therefore u is a minimizer for I and (4-3) must hold true. The same is true for the converse. Finally, we give some information about the support of the parametrized measure mllllmlzer. Corollary 4.7

supp (vx ) C {tp(x, u(x),·)

= Qtp(x, u(x),·)} ,

a.e. x E O.

3. Parametrized measures solutions of variational principles

69

Proof Observe that by the relaxation Theorem 4.1,

r iwnxN r ['P(x,u(x), A) - Q'P(x,u(x), A)] dVx(A)dx

if!

= 0,

and the integrand is nonnegative. Therefore the support of Vx should be contained where the integrand vanishes. • Let us once again emphasize the importance of understanding the restrictions that govern parametrized measures that may compete in the variational principle for 1. If one forgets this issue, the connection between both variational principles, the one for J and the one for i may be lost, and information for J may not be recovered from i if the analysis overlooks those restrictions. As a matter of fact, this is the heart of the problem of understanding relaxation and was one of our main motivations in investigating characterizations for parametrized measures generated by gradients. Let us close this chapter by looking at the one-dimensional example mentioned in the introduction. The variational principle is

J(u) =

11

[cp(u'(x))

u E H1(a, 1),

+ (u(x)

- f(x))2] dx,

u(a) = uo,u(l) = U1,

where cp(A) = (IAI - 1)2 is the usual nonconvex, double well potential and f : [a, 1] --+ R is some specific, smooth, bounded function. All the necessary hypotheses hold for p = 2. The associated functionals i and I are given by

i(v) =

.£1 [L cp(A) dVx(A) + (u(x) - f(x))2] dx,

v = {vx} xEf!'

11 L

where

u(x) = Uo

AdVy(A) dy

+ foX

=

U1 - un,

L

Advy(A)dy.

In the one-dimensional case the assumptions on the admissible JL are less restrictive since the condition that the first moment of JL be a gradient is always true. For I we get

l(u) =

11

[cp**(u'(x))

u E H1(a, 1),

+ (u(x)

u(a)

=

- f(x))2] dx,

un, u(l)

=

U1.

Observe that the second term in the integral for I is strictly convex, thus making the minimizer for I unique. Let u denote such minimizer. According to

Chapter 4. Nonconvexity and Relaxation

70

Corollary 4.6, minimizers for j are obtained by seeking the family of probability measures v such that

rp**(u'(X)) =

l

rp(A) dVx(A),

a.e. x E (0,1).

In this simplified situation it is easy to observe that given any real number

u'(x) there is a unique Vx verifying the previous condition. Indeed we can write Vx

=

{ A(X)Ol Ou'(x) ,

+ (1 -

A(X))O-l'

lu'(x) I ~ 1, lu'(x)1 2: 1,

A(X) = 1 + u'(x) 2 . This family of probability measures is the unique minimizer for I.

4. Bibliographical remarks Variational problems that lack convexity have attracted researchers over the years. This is all the more so because of the interesting applications that such analysis for nonconvex problems has. Relaxation theorems and convex envelopes in several frameworks are very well understood by now. The literature on this topic is copious. We do not claim to include all the relevant papers here. Some of them deal with different situations and need delicate techniques, especially those related to BV functions and measures. See [2], [3], [15], [54], [62], [91], [92], [94], [98], [104], [105], [137], [138], [140], [144], [152]' [194], [199], [224], [280], [293], [306], [312]. See also [197]. Many of the textbooks mentioned in Chapter 1 include some treatment of nonconvexity and relaxation. The generalized variational principle in terms of parametrized measures goes back to [314] and [315], and was described and analyzed in some detail in [77] in a framework similar to ours. Related works include [148], [179], [190], [243]. The numerical analysis of nonconvex problems has received much attention lately. Nevertheless, we lack efficient algorithms to compute oscillations. Because these take place in so small a scale, computers have a lot of trouble detecting them in an accurate way. References dealing with this topic are [57], [63], [74], [85], [147], [211], [251]' [252], [261]' [263], [281].

Chapter 5 Phase Transitions and Microstructure

1. Two main examples from continuum mechanics

We have tried to emphasize in the previous chapter the importance of the study of variational principles for which some lack of convexity leads one to consider the behavior of minimizing sequences. From the mathematical point of view, there are two ways to proceed whenever there are no minimizers as a consequence of this lack of convexity. One is to "convexify" the energy density itself or the nonconvex constraints involved in order to obtain a new functional which can be analyzed through the techniques dicussed in Chapter 3. The task is to relate the information concerning this convexified functional with the original one. Relaxation theorems refer to this issue. Another possibility is to enlarge sufficiently the class of competing objects in some kind of generalized variational setting as to include minimizers. These generalized objects are parametrized measures. They were introduced by Young in this same context to understand ill-posed variational problems. The type of oscillatory phenomena described by means of the one-dimensional example in Chapter 4 is also present in martensitic transformations where the oscillations of the deformation gradient, in the context of nonlinear elasticity, remain finite in amplitude but take place in smaller and smaller spatial scales. This extremely fine structure of alternate layers has been referred to as microstructure, a term that accepts many different meanings but intuitively reflects the behavior of minimizing sequences. These models are placed in the framework of nonlinear elasticity and the connection between continuum models and crystallographic properties of materials is made through the Cauchy-Born rule that postulates the existence of a continuous, nonnegative energy density, i.p, that provides a measure of energy corresponding to a deformed crystal lattice. The basic axiom of elasticity theory (see Section 5, Chapter 3) is that the total free energy can be represented as the integral over the reference configuration n of the local density associated with a deformation of the body u: n ---; R 3 , J(u) =

10 i.p(\1u) dx.

P. Pedregal, Parametrized Measures and Variational Principles © Birkhäuser Verlag 1997

72

Chapter 5. Phase Transitions and Microstructure

\7 U is the deformation gradient and represents a measure of the local strain around each point x E n (for the purpose of this discussion, temperature is assumed to be held constant). From the physical point of view, i.p should incorporate frame indifference and reflect material symmetry as well. These facts get translated into the invariance

i.p(QFH) = i.p(F),

Q

E

50(3),H

E

P,

that arise from the Cauchy-Born rule, where P is a set of matrices reflecting the crystalline symmetry of the material (in situations of interest P is a finite group of matrices reflecting the symmetries of one of the phases taken as reference). The consequences of (5-1) are crucial, namely, that invariance is responsible for lack of quasi convexity for i.p and ultimately for the presence of microstructure in this type of problem. Indeed, suppose that for a particular matrix F with zero energy, i.p(F) = 0, we can find Q E 50(3) and H E P such that F and QFH are rank-one related

F-QFH=ac>9n.

(5-2)

By the invariance (5-1), i.p(QFH) = 0 and were i.p rank-one convex, we would have (bearing in mind that i.p :::: 0) that i.p vanishes along the segment joining F and QF H. This means that any convex combination of F and QF H has zero energy and hence it should be contained in the zero set of i.p. In the situation we are discussing on martensitic transformations this is not so and hence i.p cannot be quasiconvex. The zero set of i.p plays a fundamental role since we can look for minimizers or minimizing sequences whose gradients take on values in this set as often as possible. If (5-2) holds, by the basic construction described in Chapter 1 related to rank-one convexity, for any t E (0,1) we can find a sequence of Lipschitz deformations, {Uj}, such that Uj -Uy E w~'(X)(n) where uy(x) = Yx is affine, Y = tF + (1 - t)QFH and \7Uj takes on the values F and QF H in alternate layers with normal n and relative frecuency t and 1 - t, respectively. This sequence of deformations is minimizing, I(uj) -+ 0, and represents a stress-free microstructure. (5-2) is the basic equation of the crystallographic theory of martensite and it can be derived rigorously from energy considerations. The invariance (5-1) gives a lot of information about the structure of the zero set of i.p. The typical situation is the following. The set P is a discrete group of several matrices accounting for the symmetry of the material we are working with. Assume that we take a particular affine, homogeneous deformation with minimum energy as a reference so that i.p(l) = 0 where 1 is the identity matrix. For each lH, H E P, we have a potential well {QlH: Q E 50(3)} made up of minimum energy matrices, each one a copy of 50(3). Altogether we obtain a finite number of potential wells which contain no segment. Under these circumstances minimizing sequences for the internal energy functional will develop oscillations taking place in a very fine scale as announced.

1. Two main examples from continuum mechanics

73

A different source of nonconvexity may be located on the set of competing functions in a particular variational principle, so that even if the functional itself is convex, the analysis leads one to consider some kind of relaxed formulation. One such interesting example that we will analyze is some detail through divergence-free parametrized measures comes from the theory of micromagnetics. Micromagnetics is a mathematical model of ferromagnetism intended to provide a description of the magnetization of a ferromagnetic body under the action of an external applied field. The theory has evolved to seek an explanation for the fine structures observed in experiments. We will restrict our attention to the rigid case in which the only state variable is the magnetization m, assumed to be a vector field over the body. One interesting assumption is that the magnetization field is assumed to be of constant length if we do not allow temperature variations. For simplicity we set Iml = 1. This hypothesis reflects the local saturation of the material. The variational principle governing equilibrium configurations of a large body for the magnetization m consists of several terms which give rise to the following energy functional

I(m) =

r
in

2

dx -

rH· mdx,

in

(5-3)

subject to the constraints Iml = 1, a.e. x E n, div(-V'u+mxn)=O inH-I(RN). Notice that the solution operator to the differential constraint, which can be solved by means of the Lax-Milgram theorem, is weak continuous from L2(n) to HI (n). The three terms in the energy functional correspond respectively to the anisotropy energy accounting for the tendency of the material to align the magnetization with specific crystallographic directions, the magnetostatic energy and the interaction energy due to the presence of an external, constant applied field H. Two cases are of particular relevance concerning the anisotropy energy density
Such a field m, were it to exist, is said to "attain" the minimum energy. We will be mainly considering conditions under which non-attainment of minima occurs and provide some insight on how the behavior of such materials can be analyzed. Suppose we were to use the direct method to look for minimizers of (5-3) and (5-4). We would find a minimizing sequence, {mk}. Because of the admissibility condition Imk(x)1 = 1 a.e. x E n, we could choose a subsequence,

74

Chapter 5. Phase Transitions and Microstructure

not relabeled, converging weakly to some m E LOO(O). This m would not be admissible unless {mk} converges strongly in LOO(O) to m. Said differently, if we insist on having Im(x) I = 1 a.e. x E 0 then {mk} must converge strongly. Roughly speaking, this is due to the fact that the unit sphere contains no segment. There is no a priori reason why to expect that minimizing sequences converge strongly, and in fact they do not. The direct method is useless in this framework. This does not imply, however, that the variational principle (5-3) does not admit minimizers. In some cases the minimum energy is attained. On the other hand, there are several places where one can find explicit examples in which minimizing sequences develop finer and finer oscillations. They certainly converge only in the weak sense so that weak limits cannot represent physical magnetizations under the saturation constraint Iml = 1, as indicated. This nonconvex constraint pushes us to consider some kind of relaxed formulation in terms of parametrized measures associated to sequences of magnetizations. Notice that the convexity or nonconvexity of cp does not playa role here. 2. Phase transitions and microstructure One of the most interesting and fascinating examples of nonconvexity in nonlinear elasticity is the case of solid-solid phase transformations in ordered materials. The behavior of such materials can be understood and analyzed as an energy minimization process. The main feature of these materials is an extremely fine mixture of the phases. This property is a consequence of the lack of "convexity" of the energy density, and this in turn comes from symmetry considerations. In order to understand the mathematical analysis behind this phenomenon, we do not need to know the specific data for the materials under analysis but rather the key facts responsible for its qualitative behavior. Assume we are dealing with a crystal considered as an elastic material. We postulate the existence of a nonnegative, free energy cp that depends on the change of shape (the deformation gradient) and on temperature as well. This is, more or less, the main consequence of the Cauchy-Born rule mentioned earlier. Above a certain critical temperature 80 , there is a stable phase, taken as reference. By stable we simply mean that it minimizes cp, so that cpe(l) = 0, where 1 is the identity matrix and 8 > 80 , At the transition temperature 80 there is a change of stability or of crystal structure, so that below 80 the stable phase is not represented by 1 any more but by some other matrix Uo describing the change in crystal structure that has taken place. Thus CPe(Uo) = 0 but cpe(l) > 0 for 8 < 80 . At the transition temperature 8 = 80 both phases may coexist CPe o (Uo) = CPe o (1) = O. Since the energy density cP = CPe o must be frame indiferent

cp(RF) = cp(F) for any rotation R. This implies that cp(R) = cp(RUo) = 0 for all rotations R. We also have to take into account that the phase represented by 1 may

2. Phase transitions and microstructure

75

have some symmetry. In fact, there is a group of rotations P of finite order representing the symmetry of the phase taken as reference. For example, if the atoms for the reference phase aligned themselves on cubic cells, then P would consists of the 24 rotations that leave a cube invariant. We should have at the critical temperature for any REP. Altogether we have found many matrices for which the free energy density cp vanishes:

where P = {I, R 1 , R 2 , ••• , Rn} and R is any rotation. We call each one of the sets {RRiUoRT : R any rotation} a potential well associated to Ui = RiUoRT and make the further assumption that the free energy density cp is positive outside the set of the wells: the zero set for cp, {cp = O}, is exactly the set of the wells. Under these circumstances, we are looking for minimizers of the energy functional

J(u) =

In cp(\lu) dx,

among all deformations u of the reference configuration 0 C R3 satisfying appropriate boundary conditions. We have already explained why symmetry considerations lead to conclude that cp can~ot be quasiconvex, and hence existence of minimizers is not guaranteed, at least it cannot be achieved by the direct method. In fact, fine phase mixtures provide minimizing sequences for which the weak limit is not a minimizer. We can consider the setting described in the previous chapter where we would allow gradient parametrized measures to compete in the minimization process. Assume that some Uo E W1,P(O) determines the prescribed boundary values we are interested in. Our generalized variational principle would read

i(v) = \lu(x) =

1

1

Jo.r

M3X3

cp(A) dVx(A) dx

M3x3

Advx(A),

u - Uo E

W~'P(O),

where v = {vxLEf! is a Wl,P-parametrized measure and we assume p to be large enough. Under suitable bounds for cp there are solutions to this variational problem. The structure of the zero set for cp plays a fundamental role if we are interested in finding equilibrium configurations, including parametrized measures minimizers, with zero energy. It is trivial to realize that this amounts to finding gradient parametrized measures whose support is contained in the set

76

Chapter 5. Phase Transitions and Microstructure

of wells for <po It is of paramount importance to pay close attention to the fact that admissible parametrized measures for j must be generated by gradients. Otherwise the variational principle would not have any significance with respect to the physical situation that we propose to examine. In obtaining useful information about the fact that parametrized measures should come from gradients, the so-called minor relations have been successfully exploited in many situations. These are nothing more than Jensen's inequality for the quasi affine functions: the minors. Because these minors are continuous with respect to weak convergence in Sobolev spaces, Jensen's inequality is in fact an equality

M(V'u(x)) =

r

JM3X3

M(A) dvx(A).

This has to hold for every minor M and a.e. x E n if v = {vx } xEn is admissible for 1. These relations yield important information in the form of necessary conditions that in some cases are sufficient to find gradient parametrized measures supported on the set of wells for <po For instance, by using the minor relations one can conclude that the existence of nontrivial microstructures is only possible if the wells are compatible. In the case of two wells this means that two matrices, one of each well, should differ by a rank-one matrix. If the energy density only has one well then there cannot be non-trivial microstructures. The structure of the set of such gradient parametrized measures is extremely complex even for just two wells, and the problem is still far from being completely solved. Most of the microstructures observed in experiments and obtained analytically apparently correspond to "laminate" structures. These will be examined in great detail in Chapter 9 in connection with the rank-one convexity condition for the vector case. These structures are a special case of gradient parametrized measures that admit a nice analytical treatment. They can be characterized by means of Jensen's inequality for rank-one convex functions. 3. The two-well problem

An interesting way of clarifying precise some of the ideas described in the previous sections and of demonstrating how complicated some issues can be concerning the full description of microstructures is to study the two-well problem where we assume that the zero set of the energy density
J(u) =

In


u = Uo on

an,

77

3. The two-well problem

where Uo is some prescribed Lipschitz deformation with finite energy I(uo) < 00. Our main assumption here is that the free energy density 'P is nonnegative, 'P(Y) = +00 if det Y :s; 0 and

K = {'P = O} = 80(2)Fl U 80(2)F2'

det Fi > O.

We have already noted the fact that under these assumptions 'P cannot be quasiconvex and we cannot expect classical minimizers. For this reason we concentrate on the generalized equivalent variational principle

J(v) =

in L

'P(A) dVx(A) dx

where v = {vX}xErl must be a gradient parametrized measure (a microstructure) with compact support (we restrict attention to p = 00 in the context of Chapters 4 and 8) and u ~ Uo E W~,OO(n) where

Vu(x) =

L

Advx(A).

We are interested in finding stress-free microstructures: J(v) = O. Obviously this is equivalent to actually having supp (v x ) C K and this condition in turn imposes restrictions on the set K itself, and on the possible boundary values Uo that may support nontrivial, stress-free microstructures. We take the term microstructure here as equivalent to gradient parametrized measure. We would like to draw some conclusions regarding three issues raised in the above paragraph: 1. Conditions on the two wells to ensure the existence of nontrivial, stressfree microstructures. 2. Affine boundary values that may support such microstructures. 3. Examples of stress-free microstructures. Underlying our analysis is the need to understand the convex hull of the set 80(2). Indeed, it consists of all matrices P of the form

p=(a

~(3

(3) '

(5-5)

a

which is elementary to verify. Furthermore, if JL is a probability measure on 80(2),

p=

r

QdJL(Q)

iSO(2)

is in the convex hull of 80(2). If det P = 1 then P E 80(2) and JL = 8p . The main tool in deriving necessary conditions in this context is the minor relation det

(L Adv(A)) =.[ detAdv(A)

78

Chapter 5. Phase Transitions and Microstructure

which should be valid whenever v is a gradient parametrized measure. This equality is also true for 1/I(A) = det(A - F) for a fixed matrix F because 1/1 is also a weak continuous function. This fact can also be proved by using the formula that follows which will playa role in some proofs. It is only valid for 2 X 2 matrices det(A - B) = det A - (adj A)T . B

+ det B,

(5-6)

where A adj A = det A 1. Notice that (adj Af . B is a linear function on the entries of A. Before going any further, we would like to consider if it is possible to have nontrivial gradient parametrized measures supported in a single well SO(2)F. This is impossible due to the weak continuity of the determinant: if v is a gradient parametrized measure supported in SO(2)F, det (

r

} SO(2)F

A dV(A)) =

r

} SO(2)F

det A dv(A).

The right-hand side is det F and the left-hand side can be written det(P F) where P E co(SO(2)). Hence det P = 1 and by the observation above this implies that v has to be trivial, a delta measure. A second easy preliminary step is to consider gradient parametrized measures supported in just two matrices, Fl and F 2 • In this case any probability measure can be written

Again, by the weak continuity of the determinant,

The left-hand side can be decomposed as

This formula is also valid only for 2 X 2 matrices. This condition clearly implies det(Fl - F2) = 0 and thus Fl - F2 must be a rank-one matrix. Otherwise v must be a Dirac mass. We now get fully into the two-well problem, and treat in succession the three issues mentioned above. 1. Our main result concerning restrictions on the set of two wells is the following. We say that the wells SO(2)Fl and SO(2)F2 are incompatible if Fl - QF2 is never a rank-one matrix for all rotations Q.

79

3. The two-well problem

Theorem 5.1

Let v be a homogeneous gradient parametrized measure with

suppv C SO(2)Fl U 50(2)F2'

det Fi

> 0,

i = 1,2.

If the wells 50(2)Fl and 50(2)F2 are not compatible, v = I5 QHi is a Dirac mass. A crucial technical fact in the proof is the next lemma.

Lemma 5.2 Let A be a matrix such that det(A - Q) > 0 for all rotations Q E 50(2). Then det(A - P) > 0 for every P E co(50(2)). Proof. Write

(3) 2 2 a Q = ( -{-J a ,a + f3 = 1. After some algebra

If det( A - Q) > 0 for all (0:, (3) in the unit circle, this means that the unit circle does not meet the circle centered at

with radius

By continuity, this last circle docs not llleet the solid unit circle either. This is the conclusion of the lemma. • Proof of Them'em 5.1. Set v ~ (1 - A)V 1 + AV 2 ,

Pi =

(r. .I

SO(2)F;

F

supp (vi) C 50(2)Fi'

i

= 1,2,

AdVi(A)) F,-l E co(50(2)),

i

= 1,2,

= /' A d/J(A) = (1 - A)P[ FJ + AP2 F2 . .If{

80

Chapter 5. Phase Transitions and Microstructure

Consider the weak continuous function

On the one hand, by direct substitution (5~7)

Due to the weak continuity, and by

1j;(F) =(1 - >.)

+ >.

r

r

(5~6)

det(A - F2P2) dvI(A)

JSO(2)Fl

det(A - F2 P2 ) dv 2 (A) JSO(2)F2 =(1- >.) (det(FI) + det(P2F2) - (adj (PIFI )? . (P2F2))

+ >. (det(F2 ) + det(P2F2) -

(adj (P2F2)? . (P2F2))

=(1- >.) (det(FI) - det(PIFI ) + det(PIFl + >. (det(F2) - det(P2F2)).

-

P2 F2 ))

Therefore we obtain the equality

(1 - >.? det(PIFl

-

P2 F 2 ) =(1 - >.) (1 - det(PI )) det(FI)

+ >. (1 - det(P2)) det(F2) + (1 - >.) det(PIFl - P2 F 2 ),

or

(1 - >.) (1 - det(H)) det(FI ) + >. (1 - det(P2)) det(F2) + >'(1 - >.) det (PI FI - P2F2) = o.

(5~8)

Assume that>. E (0,1) and the wells are incompatible, so that det(RFl QF2 ) > 0 for all rotations Q and R. Multiplying by F2~1 to the right and letting A = RFIF2~1 we have det(A - Q) > 0 for all rotations Q. By Lemma 5.2, det(A-P) > 0 for all P in the convex hull. This is equivalent to det(RFI P F2) > 0 for all such P. In particular det (RFI - P2F2) > 0 for all rotations R. Therefore (5~9) det(A - P2F2) dvI(A) > O. JSO(2)F1

r

By the formula used above

1j;(F) =(1- >.)

+ >.

r

r

JSO(2)F1

JSO(2)F2

det(A - F2P2) dvI(A)

det(A - F 2 P2 ) dv 2 (A).

3. The two-well problem

81

The first term of the right-hand side is positive by (5-9) and the second term, by the computations made earlier, is equal to

A(1 - det(P2)) det(F2) which is nonnegative (recall 1 - det(Pi ) 2': 0). Hence 'l/J(F) > 0 and by (5-7), det(P1F1 - P2F2) > O. This is a clear contradiction of (5-8) because the sum of three nonnegative terms vanishes only if each one vanishes individually. The conclusion is that if the wells are incompatible, then either A = 0 or A = 1 and in this case the probability measure is trivial (case of one well). • 2. We would like to characterize the affine boundary conditions uo(x) = Fx, F E M, that may support nontrivial, stress-free microstructures. We assume accordingly that the two wells are compatible. After an appropriate change of coordinates we can take

K = SO(2)Fo U SO(2)Fo-1,

Ff1 = 1 ± 8e1 ® e2,

where 8 > 0 is a fixed parameter and ei is the canonical basis for R2. If v is a homogeneous gradient parametrized measure, we write and hence

F

=

V=(1-A)V 1 +'\v 2,

1

Adv(A)

(1 - ,\)P1Fo + '\P2 Fo- 1,

=

where Pi E co(SO(2)) and

Pi =

r

JSO(2)

Adv i (A)Fi- 1 =

We have kept the notation F1 expressions into F, F

=

(~/3i., 0:, /3i), 0:; + /3; : : ; 1.

Fo, F2

=

FO- 1 for convenience. Placing these

= (1 _,\) ( 0:1 -/31

and for C = FTF, the Cauchy-Green tensor, write

C = FT F =

(Cl1 C21

C12), C22

F = (F(l) F(2)) .

Then we have the inequalities Cll

=

C22 =

IF(1)1 2 ::::;(1-,\) (ooi + /3i) + A (oo~ + /3~) : : ; 1, IF(2f ::::;(1 - A) 1(/31, (01) + 8(001, -/31)1 2 + A 1(/32,0:2) + 8(-0:2,/32)1 2 =(1 - '\)(1 =(1

+ 82 ).

+ 82 ) + A(1 + 82 )

Chapter 5. Phase Transitions and Microstructure

82

On the other hand by the weak continuity of det,

detF

=

i

detAdv(A)

=1

so that and consequently In the

Cll -C22

plane we have found the constraints

These determine a region D easy to draw. The question is: does every point in D come from the Cauchy- Green tensor corresponding to a gradient parametrized measure v supported in K? The answer is yes. To understand this we need to review briefly how laminates supported in four matrices can be easily constructed. For a complete discussion on laminates and gradient parametrized measures, refer to Chapter 9. With four matrices, A, B, C, D, the compatibility conditions we need in order to have a laminate are

rank(A-B) = 1, rank(C-D) = 1, rank ((AA + (1 - A) B) - (aC + (1 - a)D)) = 1, for some A, a E (0,1). In this case, any convex combination of AbA + (1- A)bB and a/jc + (1 - a)/jD will be a gradient parametrized measure (a laminate), using again the idea of layers within layers to find the corresponding sequence of gradients (Chapter 9). Let v be the laminate supported in K

(Fa and Fa l are rank-one related). For this v, F=

(1o

/j -

2Ab)

1

and the corresponding Cauchy-Green tensor

'

3. The two-well problem

83

°

As>" moves from to 1, C22 = 1 + 82 (1 - 2>..)2 goes down from 1 + 82 to 1 and then back to 1 + 82 , while Cll stays constant at l. There is another matrix Q8 E SO(2) with the property that Q8Fo is rank-one related to FO-I. Namely, after some computations, (5-10) The matrix Q8Fa is called the reciprocal twin of Fa-I. Thus we may consider the laminate and find

In this case one obtains ell

2 (2)''-1) 2) , = - -12 ( 1+8 1+8

so that as >.. runs through [0, 1], C22 is fixed at 1 + 82 but Cll goes from 1 to 1/(1 + 82) and back to l. These very same computations show that for a given>.. E [0,1] and Qp,) = Q8(1-2),) , given by (5-10) with 8(1 - 2>..) replacing 8, the matrix

is the reciprocal twin of

because (1 - >..)Fa + >"FO-

I

is a matrix of the same type as Fa. For

we reach eventually every point in D as (0", >..) E [0,1] x [0,1]' for>.. lets us move up and down and 0" from left to right. This F corresponds to the measure l/

= (1

- 0")(1 - >")8Q (A)Fo

+ (1 -

0")>"8Q (A) F-1 0

where

(5-11)

84

Chapter 5. Phase Transitions and Microstructure

This probability measure is a laminate because the rotation QUI) was so determined. 3. The next step is to study, for each possible F whose Cauchy-Green tensor lies in D, the set of gradient parametrized measures supported in the two wells with such an underlying deformation, or at least to say something about the structure or the complexity of that set. As we will shortly see, this is a much harder problem that cannot be solved completely except for some special matrices. Suppose that v = {vx}xEn is a nonhomogeneous, gradient parametrized measure supported in K where we take again Fl = Fo, F2 = F O- I : Vx = (1 - ),(x)) v; + ),(x)v~. Denote by y(x) the deformation underlying v, that is,

'Vy(x) =

L

Advx

= (1- ),(x))

r

Qdv;(Q)

i SO(2)Fl

= (1 - ),(x)) PI (x) Fl

where

Pi (x) =

+ ),(x)

r

i SO(2)F2

Qdv~(Q)

(5-12)

+ ),(x)P2(x) F2

r

iSO(2)Fi

Qdv~(Q)Fi-l,

i=1,2,

belong to the convex hull of 50(2). We have the following uniqueness result. Suppose that y(x) satisfies

Theorem 5.3

y(x) = Fx = (1 - ())FIx + ()F2x, for some (), 0

< () < 1.

X

E a~,

Then

Vx

=

(1 - ())8F1

+ ()8 F2 ,

for x E

0..

Proof Assuming that 10.1 = 1, by the divergence theorem,

F

=

=

l

'Vy(x) dx

r(1- ),(x)) r

in

r

i SO(2)Fl

Qdv; dx +

r),(x) r

in

Qdv; dx

i SO(2)F2

r

Q (1 - ),(~)) dv; dx +),* QA(~) dV; dx inXSO(2)Fl 1 - ), inxSO(2)F2 ), = (1- ),*) MIFI +),* M 2 F2 ,

=

where

(1- ),*)

),*

is the average of), over 0.. Now

(1-),(x))d Id 1-),*

Vx

x

and

),(x) d

Y

2

Vx

d

x

3. The two-well problem

85

are probability measures, and hence reduce to Dirac masses if the Mi are rotations. Furthermore, the Mi are averages of rotations, and hence lie in the convex hull of 50(2). We now have the equation

Multiplying to the right by F 1-

1

= F O- 1 = F 2 ,

(1 - 0)1 + OH = (1 - ),*) M1 +),* M 2 H, where H

= (Fo-1)2 = 1 + tel

@

(~ ~t) =

(~J1 ~~) +),* (~J2 _aJ:t1~2)·

e2, t =

(5-13)

-215. Say that

Then

Now

(1-),*)

lail ::; 1, and

implies

ai =

1. Next

implies Ih = O. Finally,

can only happen if

implies 0 =

),*.

/31

=

0, and likewise

Consequently the matrices Mi

= 1,

v~

= 151 , i = 1,2, and

We need now to show that ),(x) is actually a constant function. First, using the mixed second partial derivatives in (5-12) with Pi = 1, we conclude that ),(x) is a function of X2 alone. Then

Applying the boundary condition, we see that (5-12), we obtain ),(x) == O.

!(X2) = OtX2, and going back to •

Chapter 5. Phase Transitions and Microstructure

86

This uniqueness result is very special. Indeed for most of the matrices that may support nontrivial microstructures such uniqueness fails drastically: there even exist continuously distributed gradient parametrized measures supported in the two wells. The construction that follows is based on two of the main facts shown in Chapter 8 and stated in Chapter 1: i) the process of averaging when we have affine boundary values, Theorem 8.1; and ii) the decoupling in rank-one compatibility and oscillatory properties of nonhomogeneous gradient parametrized measures, the characterization theorem, Theorem 8.16. Let us take n = [0,1] x [0,1], and let y : n --+ R2 be a deformation with some affine boundary condition. Assume that we can actually find y with the property that F(x) = 'V'y(x) admits the decomposition

F(x) =P(x) [J1(x)>.(x)Q(>.(x))

+ P(x) [J1(x) (1 -

+ (1 -

J1(x)) (1 - >.(x)) 1] Fo

>.(x)) Q(>.(x)) + (1 - J1(x)) >,(x)l] F O- I ,

(5-14)

where we are using the same notation as in the previous section, Fot l = 1 ± 8eI ® e2, 8 > 0, >. and J1 are nonconstant, continuous functions with values in [0,1], P : n --+ 80(2) and Q(>.(x)) E 80(2) given by (5-11) is such that det {Q(>.(x)) [>.(x)Fo + (1- >.(x)) FO-I] - [>.(x)FO- I We claim that the family of probability measures v

=

+ (1- >.(x)) Fo]} = 0. {vx } xEO given by

Vx = [J1(x)>.(x)8p(X)Q(A(X))Fo + (1 - J1(x)) (1 - >.(x)) 8p(X)Fo]

+

[J1(x) (1 - >.(x)) 8p(X)Q(A(X))Fo- 1 + (1 - J1(x)) >.(x)8p(X)Fo-1] ,

is a gradient parametrized measure. This is a direct consequence of Theorem 8.16 above since by the preceding discussion, each Vx is a laminate supported in K. Therefore under the assumption (5-14) our claim is true. Let us look at the average of such v, fJ. According to the average formula, for a continuous function 'ljJ,

Ix

In Ix = In

'ljJ(A) dJ; =

'ljJ(A) dVx(A) dx

[J1(x)>'(x)'ljJ (P(x)Q(>.(x))Fo)

+ (1 - J1(x)) (1 - >.(x)) 'ljJ (P(x)Fo) + J1(x) (1 - >.(x)) 'ljJ (P(x)Q(>.(x))FO- I ) + (1-J1(x))>,(x)'ljJ(P(x)Fol )] dx.

3. The two-well problem

87

If A(X) is a continuous, nonconstant function, either {P(x)Q(A(x))FoLEf! or {P(x)FoLEO is a continuous distribution on the well corresponding to Fo. Since the density functions J.L(X)A(X) and (1 - J.L(x)) (1 - A(X)) are both nonnegative and nonconstant, the equality

iK 'l/J(A) dDI(A) = in [J.L(x)>,(x)'l/J (P(x)Q(>.(x))Fo)

+ (1 - J.L(x)) (1 - >,(x)) 'l/J (P(x)Fo)] dx asserts that VI is a continuous distribution on SO(2)Fo. The same argument is valid for the well SO(2)Fo-1. Let us find a function y : n = [0,1]2 --t R2 for which the decomposition (5-14) can be achieved. We know that this decomposition is possible if for C = FT F = VyTVy, we have the constraints CllC22 - ci2 = 1,

1 -::; C22 -::; 1 + 82 , 1 1 + 82 -::; ell -::; 1,

where as before

First of all, a map 'P of type

(r,8) ~ECl([O,l]),

--t

(r,8 + E

~(r)),

~(0)=~(1)=0,

O-::;r-::;l,

E>O,

in polar coordinates, has the properties: i) det V'P = 1, ii) 'Plr=l =id. In rectangular coordinates 'P = ('PI(Xl,X2),'P2(Xl,X2)), and it is elementary to find 'PI

= Xl COS(E ~(r)) - X2 sin(E ~(r)),

'P2 = X2 COS(E

~(r))

+ Xl sin(E ~(r)).

Direct computation yields

( O'PI ) 2 + (O'P2) 2 = 1 + xi (E( (r)) 2 _ 2E(( r) Xl X2 , &1 &1 r r=

JXI +X~,

Chapter 5. Phase Transitions and Microstructure

88

and something similar for

The point is that these two expressions, that represent the diagonal of the Cauchy-Green tensor of the deformation cP, are nonconstant in any sub domain for almost any choice of ~ (take for instance ~(r) = r(l- r)). According to our discussion, this in turn ensures that >. and (J are nonconstant functions. Given a E (0,1), consider now CPa, a variant of cP itself, (r, e)

(r, e+ aE

--+

~ (~)),

°~

r

~ a,

and extend it by the identity to the box

After a translation, let

denote the corresponding map

Ua

Ua :

Oa = [0, a] x

[o,~]

--+

Oa,

det Y'u a = 1,

ual ao " = When

E

= 0,

( au~)2 + (au;)2 = 1, aXi

Therefore we can fix E(a) 1

J 1 + 82 (Recall that

F;f

< -

i = 1,2.

aXi

> 0, sufficiently small so that

(aU1)2 + (aU2)2 <~ a

a

aXi

aXi

= 1 ± 8e1 ® e2, 8

y for Ha and

id.

,

i

= 1,2.

> 0.) Finally, let y be

= U a . Ha

= ae1 ® e1 + ~e2 ® e2,

-

:0

and a

= [0,1]2 --+ Oa, = (1 + 82 ) -1/4 < 1.

Clearly det Y'y

=1

4. An example in micro magnetics

89

so that the following inequalities are valid C11

~ (1 + 62)-1/2(1 +6 2)1/2

Cll

2'

2

0'

1

vfl+82

= 1,

1 1 + 82 '

C22 ~ (1 + 62 )1/2(1 + 62 )1/2 = 1 + 62 , C22

2'

1 0'2

1

vfl+82 =

1.

Therefore \7y admits the claimed decomposition. Moreover, in the sub domain

dt))

H;;l (Ba/2 (~, c fl, Cll and C22 are nonconstant by the computations made earlier. We have obtained a homogeneous, continuously distributed gradient parametrized measure supported on the set of the two wells.

4. An example in micromagnetics Once we know that the variational principle of micromagnetics as explained in the introduction does not lend itself directly to study by the direct method and minimizing sequences may develop oscillations, we introduce the notion of measure-valued magnetization. We are willing to accept a measure-valued solution in the sense that the oscillations described by minimizing sequenccs take place in so fine a scale that we only care about the states that participate in the oscillations and the relative volume fractions of the regions in which such states occur. These two pieces of information are contained in the parametrized measure through the support and the weight for each state in the support, respectively. Therefore we would like to let parametrized measures l/ = {l/x LE!1 compete in the variational principle (5-4). Let us consider a sequence of magnetizations, {mk}, and let l/ = {VrLE!1 be its associated parametrized measure. Because m k takes values on the unit sphere S = {Y E RN : IYI = I}, it is clear that the support of v" is contained in S for a.e ..1: E fl. Moreover,

On the other hand, if

then,

90

Chapter 5. Phase Transitions and Microstructure

and the limit of the interaction energy is

-l

H·mdx.

The magnetostatic energy, however, presents a problem when trying to identify the limit energy in terms of the parametrized measure, because the relationship between the potential u and the magnetization m is given through the differential constraint div (- V'u + mXn) = O. The clue to understanding this passage to the limit for the magnetostatic energy is the following fact. Theorem 5.4 For any sequence oEmagnetizations, {mk}, such that { divmk} is a compact set in Hl-;'~(RN), we have

uk

-->

u (strongly) in Hl(RN),

where div( -V'u k + mkXn) = 0 in H-1(R N ), m k ~ m in LOO(n), div (- V'u + mxn) = 0 in H- 1(R N ). In particular, the limit magnetostatic energy is obtained through the weak limit m in the same way that it is obtained from a genuine magnetization, provided that { divmk} is a compact set in Hl-;'~(RN). The proof of Theorem 5.4 is based on the Div-Curl lemma, a typical compensated compactness result. Div-Curl Lemma 5.5 Let 0, be a regular domain bounded or unbounded. Let {Uj } converge weakly to U in L2(n) and 10 to V in LOO(n). Suppose that { curl Uj }, { div ltj} are compact in Hl-;'~ (0,). Then Uj 10 converges weaky in the sense oE distributions to the product UV. For the proof of Theorem 5.4, apply the Div-Curl lemma to the sequences {V'u k } and {mk} for which the hypotheses of this lemma hold. Through a density argument we can obtain the convergence

{ V'u JRN

k m k Xn dx

-->

{ V'u m Xn dx. JRN

Using the differential constraint

{

JRN

V'u k m k Xn dx = {

JRN

V'ukV'u k dx

{ V'u m Xn dx = ( V'uV'udx. JRN JRN This gives us the strong convergence of the gradients in L2(n). Together with the weak convergence of the solution operator to the differential equation we get the desired strong convergence. •

4. An example in micromagnetics

91

The above considerations lead us to define a measure-valued magnetization as a family of probability measures 1/ = {I/ x } xEO whose support lies in the unit sphere S for a.e. x E n and can be generated by a sequence of classical magnetizations, mk, with {divmk} a compact set in HI~~(RN). For such a generalized magnetization 1/ = {I/ x } xEO we define its total energy as 1(1/)

=

{

(

io iRN


where m is the first moment of

1/

+ ~ { 19u1 2 dx 2

10

{ H· mdx,

io

and

The importance of having { divmk} be a compact set in HI~~(RN) consists in the fact 1(1/) = lim l(mk) k--->oo

according to Theorem 5.4. If we drop the condition on the divergences, it is always true that if 1/ = {I/ x } xEO is the parametrized measure associated to a sequence of magnetizations {mk} then

l(v) :::; lim inf 1(mk), k--->oo

(5-15)

using the weak continuity of the solution mapping for the differential equation and the convexity of the function 19u1 2 . The point is that the above inequality might be strict if we do not have some extra condition like the divergences being contained in a compact set in Hl-;'~(RN). If this condition is not assumed the energy of the parametrized measure limit of {mk} might not be the limit of the energies of {mk} and the energy for v would not have any physical relevance as indicated above. In this sense, we say that measure-valued magnetizations as defined are the ones that can be interpreted physically: they come from a sequence of classical magnetizations and their energy is precisely the limit of the energies of the magnetizations. If now A stands for the set of all measurevalued magnetizations and A, for the set of the classical ones, we have shown a relaxation result: i~f l(v) = i~f l(m). What is remarkable is the fact that the additional constraint on the divergences does not restrict further the families of probability measures in A. This is a main consequence of our analysis of divergence-free parametrized measures in Chapter 10. Specifically Theorem 10.3 establishes that

A = {v = {vx } xEO

: Vx is a probability measure and supp Vx C S, for a.e. x E n} .

Chapter 5. Phase Transitions and Microstructure

92

We can reformulate the above conclusions in the context of Section 3, Chapter 1. Let .c be

We would like to characterize parametrized measures associated to sequences in .c. If we are willing to add the condition on the divergences of mj to the definition of .c, the parametrized measures, f-.t = {f-.txLEn, associated to such sequences {(mj, V1uj)} are

where 1/ = {I/ x } xEn is the parametrized measure corresponding to the magnetizations {mj} and div(-V1u+mxn) =0,

m(x) =

f )"dl/x (>')' iRN

Since the condition on the divergences does not restrict further 1/, for our analysis we can stick to .c incorporating this compactness condition on the divergences. We would also like to understand relaxation in terms of the first moments of elements in A. Notice that these first moments are precisely the weak limits of sequences of magnetizations. Let

and for mEAl,

I**(m)

=

f
in

2

iRN

lV1uI 2 dx-

f

in

H·mdx,

(5-16)

where once again div (- V1u + mXn) = 0, and by
Theorem 5.6

{

If

infI**(m) = infI(l/) = infI(u). Ai

A

A

93

4. An example in micromagnetics

Proof Let d 1 , d and d stand for those three infima, respectively. We have already proved that d = d. We show here d l = d. By convexity, it is clear that d1 'S d since for a given TTL E A I corresponding to v E A, the last two terms in the energies are the same, and we apply Jensen's inequality to the first term. On the other hand, if rn E AI, by definition of cp** we can find Vx, a probability measure supported on S such that

(rn(x))

Is

=

and

rn(J:)

cp(A) dVx(A)

rAdvx(A),

=

.Is

for a.e. x E n. By Theorem 10.3, v = {vILEn belongs to A and comparing the • energies of v and rn we conclude d <::: d1 .

Theorem 5.7

For any cP as in the previous theorem, the infimum igf I(v) A

is attained.

Proof Let {rnk} be a mmulllzmg sequence of classical magnetizations and v = {vXLEn its corresponding parametrized measure. Again by Theorem 10.3, v is in A, and by the weak lower semicontinuity (5-15) and the previous result,

d <::: I(v)

<::: lim inf I(rn k ) = d 1 = k~x

d.

•

Thus v is a minimizer.

Corollary 5.8 i) Ifv = {vX}xEn is a minimizer for I, its first moment rn is a minimizer for I** anci

cp**(rn(x)) ii) If rn is a minimizer for [**, and

rn(x) for a.e. x

En, then v

=

= Vx

=

is

cp(A)dvx(A).

is such that (5 17) holds with

rAdvx(A),

.Is

{Vr}xEn is a minimizer for I.

(5-17)

94

Chapter 5. Phase Transitions and Microstructure

As a final observation, in the absence of an external applied field (H = 0) and if If! is a continuous, nonnegative, even function with minimum value 0, we have 1

-

d = d = d= O.

Notice that under these assumptions m = 0 is a minimizer for ]**, so that all the infima are O. This is true in particular for two important situations: the uniaxial and the cubic cases when If! is assumed to attain the minimum value along one axis or three axes.

5. Bibliographical remarks Nonconvex variational principles have been remarkably successful in the analysis of phase transitions and microstructure in crystalline materials. This energetic point of view was introduced in [34], after the pioneering work in [120][131]. Further analysis has followed and explored this perspective: [35], [42], [43], [44], [73], [77], [169], [170], [174], [181]' [195], [196], [230]. We have already indicated some general books and other references in nonlinear elasticity. See also [139]. The computational issues raised by these models are quite appealing despite their great complexity. The number of works on this topic is increasing very rapidly. Most of them focus on multiple well problems in several situations under affine boundary conditions: [63], [75], [76], [81], [82]-[88], [162]' [185], [202]' [205], [206], [253], [263]. [207] is a comprehensive survey of all these developments. The two-well problem has also been extensively studied. Most of the material in Section 3 is contained in [192]. [35], [44], [109], [230]' [231]' [244], [297], [319] are also relevant to this issue. The model of micromagnetics presented here has been taken from [171], [172], [173]. References on the foundation of micromagnetics are [59], [60], [176]. See also [175]. Some other models that differ in several aspects are considered in [55], [275], [276]. The relaxation described here for the micromagnetics functional has been examined in [108] and [306] as well. Simulation of magnetic behavior has been undertaken systematically in [182]' [183], [184]' [208], [209], [210].

Chapter 6 Parametrized Measures

1. Introduction

This chapter is devoted to general issues related to parametrized measures. For this reason it is of a technical nature. We start hy establishing a rather general existence theorem that can be applied to most of the situations one encounters in practice. This existence theorem provides a representation of weak limits, when they exist, of any composition with the sequence under consideration in terms of the parametrized measure associated to such a sequence (or possibly to some subsequence). It is important to stress that this result does not guarantee in any way that the weak limit exists. This is something to be obtained independently. If we have weak limits then they can be represented by an appropriate integral against the parametrized measure. If the weak limit does not exist (hecause of concentration effects) then parametrized measures yield a different type of information that we have yet to fully understand. Since this representation of weak limits in terms of the parametrized measure requires having first weak convergence in L1 of the sequence involved, we dedicate some effort to understanding this weak convergence and how far it is from having uniform bounds in U. This issue leads us to consider biting convergence and to pay some attention to Chacon's biting lemma. It turns out that parametrized measures always provide biting limits. Whenever this biting convergence may be improved to weak convergence, parametrized measures will represent weak limits. We also give some sufficient mnditions for this representation to be valid based on the De La Vallee-Poussin criterion of weak convergence and analyze the relationship between biting and weak convergence. We finally discuss how strong convergence becomes translated into the parametrized measure device. This is a good place to point out that parametrized measures completely miss concentration effects. They have not been designed to account for this phenomenon. This fact will become clear as the reader moves through this chapter. The phenomenon is so dramatic that two different sequences, one exhibiting concentrations and the other one with the concentrations cut off, may P. Pedregal, Parametrized Measures and Variational Principles © Birkhäuser Verlag 1997

96

Chapter 6. Parametrized Measures

share the same parametrized measure. For this reason, parametrized measures are especially well-suited to deal with bounded oscillations.

2. Existence theorem It is important for us, in preparing for the existence theorem, to understand the weak convergence in Ll (0) of uniformly bounded sequences in the Ll-norm. The property of equiintegrability was recalled in Chapter 1. Remember that a sequence of L1-functions, {fJ}, is said to be equiintegrable if for E > 0 given, one can find 8 > 0 (depending only on E) such that

for all j, if IE I < 8. The following version of this property will prove to be useful. Lemma 6.1

Let {fJ} be a bounded sequence in Ll(O),

The sequence is weakly relatively compact in Ll(O) if and only if lim (sup]

{Ifjl:::k}

j

k--+oo

Ifjl

dX)

(6-1)

= O.

Proof Notice first that

1{lfJl ~ k}1

::;]

{Ifjl~k}

1dX::;]

dx < C - inrlltl k - k'

IfJl dx

{Ifjl~k} k

<

so that for k large, I{I fJ I ~ k} I (a measurable set within bars stands for the Lebesgue measure of that set) is small uniformly in j, and by the equiintegrability property

]

{Ifjl~k}

IfJl

dx ::;

E,

for all j, and k large. Therefore the limit in (6-1) vanishes. Conversely, if the above limit is zero and E > 0 is given, we can find ko such that

]

{Ifjl~k}

IfJl

dx ::;

~, 2

2. Existence theorem for all j, and k:2 ko. Set 8 = f/(2k o). If E

I Ifjl dx =

.JB

r

.J1,.'n{lfjl<:ko}

::; ko

=

n and lEI

Ifjl dx +

< 8,

r

.JEn{lfJI?ko}

Ifjl d:r

f

lEI + 2

< ko -

c

97

f

+-2

f

2ko

f,

•

and the family of functions is equiintegrable.

In this section, we prove the basic existence theorem for parametrized measures.

The hypotheses are pretty general so that this version can be applied to many different 8ituations. Theorem 6.2 Let n c RN be a measurable set and let measurable functions such that sup .I

jg(lzJI)dx < n

Zj

:

n

--+

RTn be

(6-2)

00,

where 9 : [0,(0) --+ [0,00] is a continuous, nondecreasing function such that limt-+DO g(t) = 00. There exists a subsequence, not relabeled, and a family of probability measures, v = {v"} xESl (the associated parametrized measure) with the property that whenever the sequence {'ljJ( x, Zj (x))} is weakly convergent in Ll(n) for any CaratModory function 'ljJ(:r,>..) : n x RTn --+ R*, the weak limit is the function

"ijj(:;:)

=

r 'ljJ(x, >..) dv (>").

JRm

(6-5)

x

The idea of the proof is quite natural. The existence of v is obtained in a simple way via the compactness property for weak * topologies in the appropriate space. The more technical part of the proof requires an extension of the representation of weak limits for Caratheodory functions. Proof We divide the proof in several steps. Step 1. Existence of v. The vector space

Co(Rr71) = {f

E

C(Rm): lim f(>..) = .\-+00

o}

is a Banach space under the supremum norm. Its dual space is the space of Radon measures supported in Rm denoted M(Rm) with the dual norm of the bounded variation. Since Co (Rm) is separable, we have, according to the discussion in the Appendix, Section 7, that

Chapter 6. Parametrized Measures

98

under the duality

is

IlfLll = For each j, we define

IIj

ess sUPxEn IlfLx IIM(R"') .

IIj

E L~*(n;

M(Rm)) through the identification a E Rm. For 'IjJ E

= OZj(x) where oa is the usual Dirac mass centered at

P(!1;Co(Rm)),

('IjJ,lIj) = { ( 'IjJ(x,>.)dozj(x)(>.)dx iniR'"

= It is easy to check that

10 'IjJ(x,zj(x))dx.

IllIjll = 1,

for every j.

By the Banach-Alaouglu-Bourbaki theorem there exists some subsequence, not relabeled, and II E L~*(n;M(Rm)) such that IIj ~ II: lim { 'IjJ(x,zj(x))dx = { ( 'IjJ(X,>.)dllx(>.)dx, J-->OO in in iR'"

(6-4)

for every 'IjJ E Ll(n;Co(Rm)). Step 2. Some technical preliminaries. Let 'IjJ be a nonnegative, Caratheodory function such that {'IjJ (x, Zj (x))) converges weakly in P(!1). By Lemma 6.1, lim sup ( 'IjJ(x, Zj(x)) dx k-->oo j i{'Ij;(x,Zj(x))?k}

= O.

On the other hand since 9 is nondecreasing

g(k) sup 1{lzjl J

and limk-->oo g(k) =

00

;::: k}1 ~ sup J

( g(lzj(x)l) dx <

in

implies that lim sup 1{lzjl k-->oo j

Therefore, we can choose

mk ~

;::: k}1

= O.

k in such a way that

00,

99

2. Existence theorem Hence

ksup 1{lzjl

Finally, let

()k

2: mdl--* 0,

k

--* 00.

be auxiliary functions defined for t E R by I,

()k(t) = { 1 -It I + k, 0,

It I ~ k, k ~ It I ~ k + 1, It I 2: k + 1,

and 'lj;k(X,),) = ()k(IAI)Bk('lj;(X, A))'lj;(X, A). It is then easy to deduce the following properties: i) 'lj;k = 'lj; if 'lj; ~ k and IAI ~ k; ii) 'lj;k E Ll(O;Co(Rm)) for all k; iii) 0 ~ 'lj;k ~ 'lj; for all k; iv) {'lj;k} is a non-decreasing sequence; v) limk-->oo'lj;k = 'lj; pointwise. Step 3. Extension of (6-4). In this step we would like to conclude that (6-4) is true under the assumptions in step 2. To this end, let

We have the following estimates

i"Yj,kl

~C ~C ~C

r

} {Izj I2mk }u{ ,p(x,Zj (x )):;,omk}

r

} {Izj I :;,omk }u{1jJ(x,Zj (x)):;,ok}

r

'lj;(X, Zj(x)) dx

'lj;(x, Zj(x)) dx

'lj;(X, Zj(X)) dx

J{,p(x,Zj(x)):;,ok}

+C

~ CSUp j

r

} {Izj I:;,omk }n{ ,p(x,Zj (x)) Sk}

r

'lj;(x,Zj(x))dx

'lj;(X,Zj(x))dx

J{,p(x,Zj(x)):;,ok}

+ Cksup 1{IZjl : : : mdl· j

By the discussion in step 2, we can conclude that

100

Chapter 6. Parametrized Measures

uniformly in j. In particular, this fact implies (elementary exercise) that lim lim

J-+OO k-+oo

r 1/;mk(X, Zj(x)) dx =

lim lim

In

k-+oo J-+OO

r 1jrk(x,zj(x))dx.

In

Since 1/;mk E Ll(O;Co(Rm)) for all k, by (6-4), lim J-+OO

r1/;(X, Zj(x)) dx

In

=

lim k-+oo

r r 1/;mk(X,A)dvx()\)dx

In JRm

and by the monotone convergence theorem in the second term (using iv) in step 2) we can conclude lim J-+DO

r1/;(x,zj(x))dx InrJRmr 1/;(X, A) dVx(A) dx. =

In

Step 4. Conclusion. If we remove the nonnegativeness condition forl/J, we can always sepa-

rate 1/; in positive and negative parts,1/;+ and 1/;- (1/;+ = sup {1/;, O}, 1/;- = sup {-1/;, O}) and apply steps 2 and 3 to these two functions, bearing in mind that the weak convergence in Ll (0) brings along the equiintegrability of the sequence {11/;(x, Zj (x)) I} and therefore the equiintegrability for 1/;+ and 1/;-. Notice that 1/; = 1/;+ - 1/;- and 11/;1 = 1/;+ + 1/;-. For ~ E LOO(O) we can take tp(x, >..) = ~(x)1/;(x, >..), so that tp is a Caratheodory function itself to which we can apply the preceding arguments. Observe that the weak convergence in Ll(O) of the sequence {1/;(x,Zj(x))} implies the same for {~(x)1/;(x,Zj(x))}. Thus (6-4) also holds for tp, and since ~ E LOO(O) is arbitrary, we obtain

1/;(x,Zj(x))

~ -:;j;(x)

=

r 1/;(x,)")dv ()")dx x

JRm

in Ll (0). Finally, it is not hard to check that almost every Vx is a probability measure. By weak lower semi continuity of the norm

Ilvll ~ l~r::~f Iloz] II =

1,

so that IlvxIIM(Rm) ~ 1 for a.e. x E o. If we take in particular 1/; = XBR(X) for BR the ball of radius R centered at the origin in (6-4), then

r r 1 dVx(>") dx = lim JI3rRnn 1 dx = IBR n 01.

J BRnn JRm Therefore

IBR n 01

J-+DO

=

~ ~

r

r

r

Ilvxll

JBRnn JRm JBRnn

1 dVx(A) dx

dx

IBRnol,

and Vx is equal to its total variation for a.e. x E 0, i.e., Vx 2: 0 and

IlvxIIM(Rm) = l.

•

2. Existence theorem

101

°

A particularly important example is obtained by taking g(t) = t P for p ~ 1 (we can also allow < p < 1). In this case,every bounded sequence in LP(D) contains a subsequence that generates a parametrized measure in the sense of Theorem 6.2. An important remark to bear in mind when working with parametrized measures is that in order to identify the parametrized measure associated to a particular sequence of functions {Zj} (obtained perhaps in some constructive way or using some scheme), it is enough to check

for every
E

Co(Rm) where as usual

It is even enough to have

lim { )->00

in

~(x)
{

in

~(x)

(..)dvx(>\)dx

i R",

(6-5)

for ~ and

for every Caratheodory function 'lj; such that {'lj;(x, Zj(x))} is weakly convergent in Ll (fl). The reason for this is that probability measures v are identified by their action on Co(Rm). (6-5) identifies each lIx for a.e. x E fl. There are two interesting situations where this remark will have some relevance for us. For reference, we include them in the following lemma. Lemma 6.3 Assume that we have two sequences, {Zj} and {Wj}, both bounded in LP(D). i) If I{Zj #- Wj} I ----; 0, the parametrized measure for both sequences is the

same. ii) If

{Zj} and {Wj} share the parametrized measure.

102

Chapter 6. Parametrized Measures

Proof The proof is simple. Let 'P E Co(Rm) and

~ E

L1(D). Then

The integrand on the right-hand side is a L1 (D)-function and it is integrated over a sequence of sets of vanishing measure. Hence the limit vanishes as j ----t 00, and this in turn implies that the weak limits for {'P(Zj)} and {'P(Wj))} are the same. By the above remark both sequences share the parametrized measure. For ii), use the dominated convergence theorem to examine the difference

•

A helpful example of this situation is the following. Assume {Zj} is uniformly bounded in LP(D) and let v = {vX}xEO be its associated parametrized measure. Consider the truncation operators

IAI ::; k, IAI > k. We claim that for any subsequence k(j) ----t 00 as j ----t 00 the parametrized measure corresponding to {Tk(j) (Zj) } is also v. To this aim, we simply notice that

1{lzjl > k(j)}I:::; if k(j)

----t

s~p J

IlzjIILP(o) k(j)P

----t

0

00.

3. Sufficient conditions for representation of weak limits A crucial point in Theorem 6.2 in order to have the representation (6-5) of weak limits in terms of integrals against the parametrized measure is to have "a priori" the weak convergence in L1 (0) of the sequence whose weak limit we are concerned about. We have also insisted upon the fact that uniform bounds in L1(D) are not sufficient to ensure this weak convergence. Equiintegrability should be taken care of. For this reason it is important to rely on criteria that under suitable hypotheses enable us to ascertain this equiintegrability. One such important weak compactness criteron in L1 (0) that can be used in some situations is the De La Valle-Poussin criterion. The sufficiency of such a result is an immediate consequence of Lemma 6.1, and this sufficiency is in fact the part of the result which is most useful.

3. Sufficient conditions for representation of weak limits

103

Lemma 6.4 (De La Vall6-Poussin criterion) Let 0 be bounded. The sequence {fj} is sequentially weakly relatively compact in L1 (0) if and only if

(6--{)) for some continuous function 'ljJ : [0,(0)

.

--->

'ljJ(A)

llIn - , A

A-----tCXl

R with

=

(6-7)

00.

For the sufficiency, let us suppose that there is a function 'ljJ satisfying (6-6) and (6-7). We want to show that (6-1) is true. For E > 0, take M such that ME:;:> C where C=sup r'ljJ(lfjl)dx
in

J

Because of (6-7), and for k large enough, MA ~ 'ljJ(A) if A:;:> k. Thus

r

i{lfjl"2k}

Ifjl dx

~

:1

r

i{lfjl"2k}

'ljJ(lfjl)dx

~

:[C ~

E,

uniformly in j. Therefore the sequence is weakly relatively compact by Lemma 6.l. In the spirit of this criterion we can give the following proposition that allows us to have the representation of weak limits in terms of parametrized measures. According to Theorem 6.2 this requires the weak convergence in L1 (0) of the sequence involved.

Proposition 6.5

Let 'ljJ : 0

X

Rm

--->

R be a Caratheodory function such that

l?b(:r, A)I ~ ¢(IAI), for a.e. x E 0, where ¢ E L~c(R). Let {Zj} be a sequence with sup J

rg(lzj I) dx

in

= C

<

00,

where 9 is a continuous, non decreasing function and lim ¢(t) i->CXl

g(t)

=

°

1imt->CXl

g(t) =

00.

If

(6-8)

then

where v = {v x L'En is the parametrized measure associated to the sequence

{Zj} (or possibly to a subsequence).

Chapter 6. Parametrized Measures

104

Proof Notice that the conclusion of the proposition is that lim

J->=

r~(x)1j;(x,zj(x))dx = inr~(x) iRrnr 1j;(x,>')dv (>\)dx

in

x

L=(0,). Since '¢ E Ll':;'c(R), choose mk

for all

(6-9)

~ E

--+ 00

such that

Then

uniformly in j, where g(t) 2: Mk,¢(t) for t 2: mk and Mk --+ 00 by (6--8). This implies the weak convergence in L1 (0,) of {1j;(x, Zj (x))} and thus the representation (6-9) holds. • A particular, important example is g(t) = t P , P > 0 and ,¢(t) = t q , p > q > o. In this case we have the representation (6-9) when the sequence {Zj} is uniformly bounded in LP(0,) and 11j;(x, >')1 :s; 1>'l q . However, Proposition 6.5 fails if p = q, so that for functions 1j; that grow like the pth power in >. the representation (6-9) may not be valid. This brings us to the question of what is the relationship between both terms in (6-9) in this situation when we do not have equality. In order to understand this question it is convenient to introduce the notion of biting convergence and compare it to weak convergence. We are going to explore this issue in subsequent sections. We close this section with a remarkable example. When equiintegrability fails, concentrations may develop even in a rather nasty way. This phenomenon is responsible for failure of the representation (6-5). Our example is one-dimensional. Consider the sequence of functions defined on 0, = (0,1) by

j(x) = {j2 /2, for x E (k(j J

0,

otherwise.

+ 1)-1 -

r 3, k(j + 1)-1 + r 3), k = 1,2, ... ,j,

4. Chacon's biting lemma and biting convergence

Then

IlfiIILl(ll) = 1 for

105

all j, and for cp continuous

r ip(x)fi(x) dx L 1 ) Jo 1

k( '+1)-'+

j

=

k-1 ·2

J

= :2

)

·2

·-3

k(j+1)-'-j-3

Lip(x) dx 2

2

j

LJ

-:;3CP(Xk)

k=l

1

j

J

k=l

=--;Lcp(xA:) --+

t cp(x) dx,

J[)

r

where the points Xk E (k(j + 1)-1 - j 3, k(j + 1)-1 + 3 ). Hence the sequence {Ij} converges weak * in the sense of measures to 1. For T fixed, if j2/2 2' T then {I fj I 2' T} = {Ij =I- O} and

J

{lfJI;:"r}

Ifj I dx

j2 2

=-

~ = 1.

2J

Therefore lim sup!

r-+oc

.j

. {lfJI;:"r}

Ifjl

dx 2' 1,

and by Lemma 6.1 the sequence cannot be weak convergent in L1(0). What is the parametrized measure associated to {Ij F We will answer this question after the discussion of the next section. Note how this example also illustrates that convergence in the sense of distributions and pointwise convergence are different. 4. Chacon's biting lemma and biting convergence Whenever a bounded sequence in L1 (n) is not equiintegrable, one can "remove" the set where concentrations occur and be left with a well-behaved sequence. This is essentially what Chacon's biting lemma says. The proof can be done in a very general and abstract setting. We restrict attention, however, to the framework in which we will be using this fact.

(Chacon's biting lemma) Let {fd be a uniformly bounded Theorem 6.6 sequence in L1 (n), sup Ilfi IILI(n) = C < 00 . .J

There exists a subsequence, not relabeled, a nonincreasing sequence of measurable sets nn CO, Innl "'" () and f E £1(0) sllch that fj ~ f

for all n.

in L 1 (n \ nn)

Chapter 6. Parametrized Measures

106 Proof. For j, kEN set

O;j,k =

Notice that the sequence

r

} {lfj I?k}

{SUPj O;j,k}

L

Ifjl

dx 2: O.

is monotone and nonincreasing. Let

= lim sup O;j,k 2: O. k-+oo

j

If L = 0, by Lemma 6.1 we can take Dn = 0 for all n because in this case weak convergence in Ll(D) holds for some subsequence. Let us just assume that L > O. For each mEN, let jm be such that

> SUp 0; . 2m .) ,

1m, 2m -

0; .

J

-

1

m.

In this way, (6-10) By monotonicity there also exists the limit lim sup

r-+oo

m

r

J{r~lfjml<2m}

Ifjm I dx = L' 2: 0,

and in particular there exists a subsequence m(r) for r sufficiently large, such that

Finally, bearing in mind the monotonicity of

{SUPj O;j,k},

The last inequality holds because of (6-10). Hence L' = 0, i.e., lim (sup

r-+oo

m

r

J{r~lfjml<2m}

Ifjm I

dX)

= O.

(6-11)

4. Chacon's biting lemma and biting convergence

107

Define the exceptional set On by

U {lfJml ~ 2m} C O. 00

On

=

m=n

We have the claimed properties: i) On+1 C On, for all n; ii) 10ni ~ 0:

and,

L 00

10nl::;

1{lfJml ~ 2m }1 ::; C

m=n

L 00

1

2m

'

m=n

This last term tends to 0 as n tends to 00. iii) For fixed n and keeping in mind (6~ 11 ) , 0::; lim sup

r->CXlm~n

1

r:"::{lfi m l}\!1 n

::; lim sup { r->oo

m~n J{r:"::lfiml<2m}

IfJm I dx Ihn I dx ::; O.

This implies, by Lemma 6.1, that the sequence {h.,,} is sequentially weakly relatively compact in Ll (0 \ On) for all n. The following technical proposition, whose proof is left as an exercise to the interested reader, helps us in concluding the proof.

Let {ij} be a sequence of functions bounded in Ll (0) such that there exist sets On C 0, On+l cOn, 10ni ~ 0 with {fJ} sequentially weakly relatively compact in Ll(O \ On) for all n. There exists a subsequence {ijk} and i E Ll (0) such that

Proposition 6.7

fJk ~ for all n.

i

in Ll(O \ On)

•

In the example given at the end of the previous section, the exceptional sets are precisely {fJ i= O} and then trivially fJ ~ 0 in Ll(O\ {ik i= O}) for all k. Even when the weak convergence in Ll (D) fails, we can still consider the function "if; as defined in (6~5). This function can no longer be the weak limit of {~(x, Zj(x))} as we have placed ourselves precisely in the situation when there is no weak limit at all. Yet, one might ask what is the relationship between "if; and {~( x, Zj (x))) if any. We are led to introduce biting convergence.

108

Chapter 6. Parametrized Measures

The sequence {fj} C L1 (0) converges in the biting sense to fELl (0) and is denoted

if there exists a nonincreasing sequence of measurable sets {On} such that 10ni "\, and

°

for all n. We may restate Chacon's biting lemma by saying that a uniformly bounded sequence in L1 (0) contains a subsequence converging in the biting sense to a function in £1 (0). The relationship between biting convergence and parametrized measures is given in the following theorem.

Theorem 6.8 Let {Zj} be a sequence of vector-valued functions with associated parametrized measure v = {vx } xEO' Ifcp : OxRm ----t R* is a Caratheodory function such that the sequence {cp(x,Zj(x))} is uniformly bounded in L1(0), then possibly for a subsequence cp(x, Zj(x))

~ 'P(x) =

r

JR'"

cp(x, A) dVx(A) dx.

(6-12)

°

Proof The proof is elementary. By Chacon's biting lemma, there exists a collection of subsets {On} and r:p E L1(0) such that 10ni "\, and

for all n. By Theorem 6.2 whenever weak convergence in Ll(E) holds for any subset E C 0, the weak limit has to be 'P in (6-12). Since 10ni "\, 0, we conclude that r:p = 'P a.e. x E O. • Parametrized measures always yield biting limits under the very mild assumption of boundedness in L1 (0). Having this remark in mind, it is easy to conclude that the parametrized measure corresponding to the one-dimensional example described in the previous section is 80, the Dirac mass centered at 0, for all x E (0,1).

5. Biting convergence and weak convergence In some circumstances biting convergence may be improved to weak convergence so that parametrized measures will provide weak limits. This amounts to discarding the possibility of concentrations. The following lemma gives a necessary and sufficient condition for such an improvement: biting convergence may become in fact weak convergence.

5. Biting convergence and weak convergence

109

Lemma 6.9 Let fJ : n ---+ R + (fj :::: 0) be a sequence of measurable functions in Ll (n), converging in the biting sense to f ELI ([2). A subsequence converges weakly in Ll([2) if and only if

1

Inr fj(x) dJ; S; nf(x) dx.

liminf ]->CXJ

Moreover, the whole sequence {J]} converges weakly in LI (12) to if lirnsup ]->oc

r fj(x) d:r Inr f(x) dx. S;

Ji:I

(6-13)

f if and only (6-14)

Pr-ooj. The proof is simple. Assuming biting convergence and the failure of the Dunford-Pettis criterion leads to the failure of (6-13). Let {nn} c n be the sequence of subsets associated with the biting convergence so that l12nl ~ 0, nn+1 c 1217 and fj ~ f in Ll(n \ 12,,)

for all n. We may assume, working with subsequences, that lim J-->(XJ

1

fj dx S;

n

Inr f dx <

00.

(6-15)

Suppose that no subsequence converges weakly in L1 ([2). By the DunfordPettis criterion, there exists E > 0 and a subseqllenr:e {k n } such that f

1

<::

On

fkn dx

for all n sufficiently large, because outside In particular, if i > Tl, since !he, :::: 0,

(<:: and for fixed

Tl,

i

r. h

.in

<::

we do have weak convergence.

r f",

.JOn

l

cix,

> n,

r.h, d.T jnn ki d.T + j =

In

:::;, E

Finally, letting i

dx

nn

+

o\n"

r

J!2\nn

hi dx.

---+ 80,

lim

rhi dJ; ::::

1.~= .In

E

hi dT

+

1.

11\n"

f

d:r.

This is truc for every n, and consequently

l-nG.lnr fk dx :::: + .Inr f dx, lill.l

contrary to (6 15).

i

f

•

Chapter 6. Parametrized Measures

110

A straightforward corollary is the following fact whose proof is left as an exercise.

Corollary 6.10 Let {Zj} be a sequence of vector valued functions with associated parametrized measure v = {vx } xEn' IEfor CPo, a nonnegative Caratheodory function, we have

then lim r cp(x, Zj(x)) dx J->OO

JE

for any measurable subset E ['PO

= r r cp(x, A) dVx(A) dx < 00,

JEJRTn

c n and

for any cp in the space

= {cp, CaratModory functions, Icpl

~

C(1

+ CPo)} .

If in spite of all efforts Corollary 6.10 cannot be applied so that concentrations

may arise, we still can draw some information that might be helpful in some circumstances. Theorem 6.11 If {Zj} is a sequence of measurable functions with associated parametrized measure v = {vX}xEn, liminf r 'lj;(x,zj(x))dx J->OO

JE

~

r r

JEJRTn

'lj;(x,A)dvx(A)dx,

(6-16)

for every nonnegative, Caratheodory function 'lj; and every measurable subset

Ecn.

Proof If the left-hand side of (6-16) is infinite, there is nothing to be proved. If it is finite, the sequence {'lj;(x, Zj (x))} is a bounded sequence in Ll (E). If we set as usual

then

'lj;(x, Zj(x)) l:."if

in Ll(E).

By Lemma 6.9, it is not possible to have the strict inequality

JEr "if(x)dx > liminf JEr'lj;(x,zj(x))dx. J->OO

•

Strict inequality in (6-16) occurs when the sequence {'lj;(x, Zj(x))} develops concentrations. In this sense we say that parametrized measures do not capture concentration effects. It is obvious that Theorem 6.11 still holds true if 'lj; is bounded from below by some constant.

6. Strong convergence

111

6. Strong convergence We would like to understand how strong convergence gets translated into the parametrized measure. A first thought is that since parametrized measures are a device to keep track of oscillations, and strong convergence rules out this phenomenon, one can expect that parametrized measures associated with strong convergent sequences are trivial. In this section we restrict attention to the case in which g(t) = tP • Proposition 6.12 Let {Zj} be a sequence in LP(o') such that {Izj jP} is weakly convergent in L1 (0,) for p < 00 and l/ = {l/x} xEn is the associated parametrized measure. Zj ---> Z strongly in LP(O,) if and only if l/x = 8z (x) for a.e. x E 0,.

Proof Let us consider the Caratheodory function 'IjJ(x, oX) = loX - z(x) IP . Because of the hypothesis on {Zj} when p < 00, the sequence {'IjJ(x, Zj (x))} is weakly convergent in L1 (0,) and therefore the integral representation in terms of v is correct lim

r

J~= in

'IjJ(x,zj(x))dx=

rr

in iR'"

'IjJ(x, oX) d8 z (x) (oX) dx =0,

whence Zj ---> Z strong in LP(o'). Conversely, if Zj ---> Z strong in LP(O,), for any continuous, bounded function 'IjJ(oX), we would have 'IjJ(Zj) ---> 'IjJ(z) strong in LP(o'). This implies, in particular, that for any measurable E c 0"

r

iE

'IjJ(z(x))dx =

rr

iE iR'"

'IjJ(oX) dvx(oX) dx.

We can conclude that

for a.e. x E 0,. The arbitrariness of'IjJ leads to l/x = 8z (x) for a.e. x E 0,.

•

The condition on the weak convergence of {lzjIP} for p < 00 is necessary as the one-dimensional example studied in Section 3 shows. Notice also that this fact is not true for p = 00. Take 0, = (0,1) and Zj = x j (jth powers) for x E (0,1). It is easy to find that l/ = 80 but {Zj} does not converge strongly to 0 in L=(o'). What at least is true is the fact that being the parametrized measure a delta prevents oscillations. It is also helpful to consider parametrized measures coming from sequences for which we have strong convergence only for some components of the sequence but not for all of them. In this case strong convergence reflects triviality of the parametrized measure for the corresponding components.

Chapter 6. Parametrized Measures

112

Proposition 6.13 Let Zj = (Uj,Vj): r! -+ Rd X R m be a bounded sequence in LP(r!) such that {Uj} converges strongly to U in LP(r!). Ifv = {vX}xEO is the parametrized measure associated with {Zj}, Vx = 8u(x) ® J.Lx a.e. x E r!, where {J.Lx} xEO is the parametrized measure corresponding to {Vj}.

Proof Let 'l/Jl functions, so that

Rd

-+

Rand 'l/J2 : Rm

'l/Jl(Uj) 'l/J2(Vj)

~ ~2(X) =

-+

R be continuous, bounded

-+

'l/Jl(U) in LP(r!),

r 'l/J2()..) dJ.Lx()..) JR"'

in U(r!),

1 1 -+-=l.

P

q

(In fact, 'l/J2(Vj) .2. ~2(X) in LOO(r!) if 'l/J2 is bounded.) In this case,

'l/Jl(Uj)'l/J2(Vj) ~ 'l/Jl(U)~2(X) for any E

c

in Ll(E)

r! (this is easy to check) and therefore

rr

JE JRdXR",

'l/Jl()..d'I/J2()..2)dvx ()..1,)..2)dx

=

rr

JE JRdxRm

'l/Jl()..1)'l/J2()..2) d(8u (x)()..d ® J.Lx()..2)) dx.

•

The arbitrariness of 'l/Jl, 'l/J2 and E proves the result.

We have already see the relevance of this proposition in dealing with variational principles (Chapters 2 and 3). 7. Appendix We need to give a few basic notions of LP-spaces when the target space for functions is some general Banach space X with dual X'. For r! C RN we write LP(r!; X) = {f : r!

-+

X: f is strongly measurable and

10 IIf(x)ll~ dx < oo}.

Such a function f is said to be strongly measurable if there exists a sequence of simple, measurable functions {h} such that h(x) -+ f(x) a.e. x E r! and

10 Ilh(x) - ik(x)ll~ dx

-+

0,

j,k

-+ 00.

We write L~(r!; X) =

{f: r!

-+

X : f is weakly measurable,

function of x and

Ilf(x)llx

10 Ilf(x)ll~ dx <

00 } .

is a measurable

8. Bibliographical remarks A function

f

113

is weakly measurable if for every T E X' the function of x,

x r--t U(x), T) is measurable. In the same way L~*(n;X') =

{f:

n -+ X': f

measurable function of x and

* measurable, Ilf(x)ll x '

LIlf(x)II~,

is weakly

dx <

is a

00 } .

LP(n; X), L~(n; X) and L~.(n; X') are Banach spaces under the LPnorm.

Theorem 6.14

Let X be a separable Banach space with dual X'. Then 1 :::; p <

under the duality

(I,g) where

f

E LP(n;

=

1

00, -

p

1

+-

q

= 1,

L

(I(x),g(x))dx

X) and 9 E L~*(n; X').

The particular case we are interested in is

X = Co(Rrn) = {f : Rrn

-+

R: lim f(A) = \->00

o} ,

X' = M(Rm) = { bounded, Radon measures on Rm}.

In this case we have the duality

8. Bibliographical remarks General treatments of parametrized measures and applications to different problems can be found in [24], [27], [46], [212], [213]. It is interesting to have a look at the original papers of Young, [314] and [315]. See also [93] and [302]. The existence theorem presented here is basically the version contained in [31], with some variants. In [132]' there is an existence theorem based on the technique of slicing measures. With regard to general books on measure theory, we have included several choices: [47], [164], [214], [283]. See also [136]. A complete analysis of the De La Valh§e-Poussin criterion for weak convergence in L1 may be found in [214]. The one-dimensional example included in Section 3 has been taken from [39], in connection with Chacon's biting lemma. This is a good reference for a more general proof of Chacon's lemma. See also [58]. The proof that has been carried

114

Chapter 6. Parametrized Measures

out here for this lemma is a less ambitious, more direct one. The book 1116] is a good source for an accesible treatment of LP-spaces when the target space is another Banach space. In particular the duality Theorem 6.14 has been taken from this reference. Biting convergence has also been considered in a different setting in [317]. Some of the ideas explored in this chapter have been also studied in [40], [191] and [260].

Chapter 7 Analysis of Parametrized Measures

1. Introduction

In this chapter we shall analyze more closely parametrized measures and introduce the basic tools to deal with these families of probability measures. Some of these will be used several times later. Our main goal here is to characterize parametrized measures: we are interested in knowing when a given family of probability measures can actually be generated as the parametrized measure by some sequence of functions. At this stage we do not place any further restriction on the sequences we would like to consider except for boundedness in some P(O). In this regard we place ourselves in the context of Section 2 of Chapter 2. As a matter of fact, the main theorem of this chapter, Theorem 7.7, can be proved directly taking advantage of the analysis carried out there and extending it to the case p finite by means of some technicalities involving truncation operators. This will actually be our approach to pass from p = 00 to finite p in Chapter 8 under the gradient constraint. Nonetheless we have chosen to proceed in a different way with the idea in mind of preparing some of the main techniques for the analysis of gradient parametrized measures pursued in Chapter 8. The conclusion here is that there is no real restriction, except for a technical assumption, on such families of probability measures: given any family II = {lI x LEn, there always exists some sequence {Zj} whose parametrized measure is precisely lI. The technical condition on II is that the function of x

should be an £1 (O)-function if we insist on having uniform LP(O) bounds on our sequence. The functions Zj are assumed to take values on Rm. If we want to restrict further the type of sequences, various things can happen. There are examples where we obtain fundamental restrictions on the parametrized measures that such sequences can generate. This is the case of gradient parametrized measures to be discussed in the next chapter. There are instances, however, where we do not find any extra constraint on such P. Pedregal, Parametrized Measures and Variational Principles © Birkhäuser Verlag 1997

Chapter 7. Analysis of Parametrized Measures

116

families of probability measures. This surprising fact occurs with divergencefree parametrized measures. They will be analyzed in Chapter 10. We need to discuss briefly some classes of functions and spaces to provide the functional analytical setting appropriate for our purposes. Let 9 : R+ ---* R+ be a nondecreasing, continuous function with limt--->CXJ g(t) = 00. Consider then u(n)

=

{z : n

---*

R m , measurable:

10 g(lz(x)l) dx < oo} .

Unless we demand more properties of g, L9(n) may not be a vector space. We do not need this structure on L9 (n). For 9 (t) = tP , P 2: 1 we recover the usual LP(n)-spaces. Similarly we set [9

= {cp : R m

---*

R, continuous . lim cp(>-) eXists} . . 1>-1--->00 1 + g(I>-I)

It is not difficult to see that [9 is a separable, Banach space under the norm

Ilcpll = II 1 +cpC) g(I·I) II Loo(R=) . As a matter of fact, [9 is isomorphic to C(K) under the sup norm where K is the one-point compactification of Rm. The dual space (£9)' strictly contains the probability measures in Rm, fJ., such that

There are some other objects in ([9)' as well. For instance (T

,cp

)

= lim

cp(>-)

IAI--->oo 1 + g(I>-I)

belongs to (£9)'. If 9 = +00 for t 2: R, then the space [9 should be changed to

[9 =

{cp : R m

---*

R* U {+oo} , continuous : cp

.

cp(A)

= +00 for 1>-1 2: Rand

.}

bm (1'1) eXists I>-I--->R 1 + 9 /\

,

but everything else is the same. This case should be considered in order to include the case L9(n) = Loo(n), but we do not need to make any distinction between these two cases in what follows.

2. Homogenization and localization

117

2. Homogenization and localization There are two elementary operations for analyzing parametrized measures: averaging and localization. Both processes consist in obtaining a homogeneous parametrized measure from one which is not. In the average or homogenization process, we try to somehow record in a single homogeneous parametrized measure all the information contained in all individual elements IIx for x E n. While in the localization procedure, by means of a usual blow-up technique, we concentrate on a particular parametrized measure lIa for a E n. We treat them succesively. The localization principle is important because it allows one to deduce properties of individual members of a family of probability measures. We will use it to derive necessary conditions in characterizing parametrized measures. For the averaging procedure, Vitali's covering lemma enabling us to have a countable, pairwise disjoint, covering collection from any covering family of subsets is crucial to our analysis. It is also a fundamental technical tool for the proofs of characterizations of parametrized measures. A discussion of it can be found in the Appendix.

Theorem 7.1 Let nand D be two regular domains in RN with lanl = O. Let {Zj} be a sequence of measurable functions over n, such that

laDI =

for g, a continuous, non decreasing, nonnegative function with limt--+oo g(t) = 00. Let II = {lIx } xE!1 be the parametrized measure associated to some subsequence, still denoted {Zj }. There exists a sequence {Wj} of measurable functions defined over D such that sup/, g(lwj(x)l)dx < 00, J

D

and its parametrized measure is D, homogeneous, given by

Proof The family of subsets of D given by

Aj =

{a +

En

cD:

aE D,

E:::;

y}

is a Vitali covering of D. There exists a countable collection {aij Eij :::; l/j, pairwise disjoint and

+ Eijn} ,

Chapter 7. Analysis of Parametrized Measures

118

Notice that

2:i E~ = IDI / Inl. Let us define Wj(X) =

if x E aij

+ Eijn.

Zj

(X

~ijaij )

By a natural change of variables

=L t

:S C

tf.; in g(lzj(Y)I) dy

IDI < w

00.

On the other hand, and using the same change of variables, if

~ E

C(D) and

r
iD

i

=

L E~ ~(aij + EiiYiJ) inr
Here we have utilized the mean value theorem for integrals. We recognize in the first term a Riemann sum for the integral of ~ in D and lim r rxo}D iD ioiR'" =

1~(x)

H

dx (v,
By the comments after the proof of Theorem 6.2, this implies that v, homogeneous, is the parametrized measure associated to {Wj}. • For the localization principle that follows the Radon Nikodym and the Lebesgue Differentiation Theorem are invoked at some point.

Theorem 7.2

Let nand D be as before. Let {Zj} be such that

C = sup J

r g(lzjl) dx <

in

00,

(7-1)

where as usual, g is a continuous, nonnegative, non decreasing function with limt->cxog(t) = 00. Let v = {vX}xEO be its parametrized measure. For a.e. a E n there exists a sequence {zj} defined on D such that sup J

r g(lzJI) dx <

iD

and its parametrized measure is

Va,

00,

homogeneous.

2. Homogenization and localization

119

Proof We use a blow-up argument around each point a E n. Condition (7-1) enables us to affirm (through the Banach-Alaouglu-Bourbaki theorem) that

in the sense of mewmrel:) lim

J~OO

1:)0

that

1~(x)g(IZjl) II

dx

=

Jor~(x) dtL(x) ,

for any continuous ~, where Ii, is a nonnegative, finite measure. If now stands for a continuous function such that

~a,p

then for some constant A1 > 0, lim sup lim sup 1'->0

j ~=

1

~

p

II

g(IZjl)Xa+pD(X) dx

-s: lim sup lim sup ~ p->()

j

->= p

. -s: hmsup N1 1'->0

-s:

p

rg(IZjl)~a,p(x)dx

J0

1 n

Xa+2pD(X) dp,(x)

dll

M~(a).

dx

By the Radon-Nykodirn theorcm, Lhe singular part of tL with respect to the Lebesgue measure is concentrated on a set of N-dimensional measure O. Therefore ¥X(a) <

00

for a.c. a E nand

lim sup lim sup p->()

~

j~oo p

.Inr g(lzJI)x(J+pD(X)dx < 00,

a.e. a En.

(7-2)

Define the functions

Z'j,p(x)

=

zj(a+ px),

x E D,p > O.

If cP E Co(R"') and ~ E L=(D), we have

1cp(zj,p(x))~(x) 1 dx

cp(zj(a + px))~(x) dx

= =

~

p

.Inrcp(Zj(Y))Xa+pD(Y) ~ (Y -p a)

dy.

120

Chapter 7. Analysis of Parametrized Measures

Passing to the limit in j first, yields .lim

r cp(zj,p(x))~(x) dx = P~ Inr CP(Y)Xa+pD(Y) ~ (Y -P a) dy,

J~OOJD

since {cp(Zj)} converges weakly in Ll(O) to

cp(y) =

cP given by

r cp(,x) dvy(,x). JRm

Next, by the Lebesgue differentiation theorem lim lim

p-+O J-+OO

r

cp(zj p(x))~(x) dx = JD '

r

lim cp(a + px)~(x) dx p~oo JD

= cp(a)

1~(x)

dx,

for a.e. a ED. Due to the separability of Co(Rm) and Ll(O), we may choose a subsequence of { zj,p}, which we call {zj}, such that

r

r

lim ~cp(zj) dx = cp(a) ~ dx, J~OO~ In for every cp E Co(Rm) and ~ E Ll(O) (by density). Since for a.e. a E 0 and by

(7-2)

sup J

JDr g(izji) dx <

00,

we conclude that the parametrized measure associated to the sequence Va, homogeneous. In both of these theorems one could take D

=

{zj}

is •

O.

3. Riemann-Lebesgue lemma The Riemann-Lebesgue lemma is one of the most interesting nontrivial examples where we can actually determine explicitly the underlying parametrized measure associated to some sequence of functions. There are a number of different versions of it. We will concentrate on the most general of all and go into more specific versions. In some sense, this lemma is a homogenization fact and its proof is contained in the homogenization theorem, Theorem 7.I. Lemma 7.3 Let 0 and D be regular domains in RN with 1001 = 10DI = 0 and f E LP(O). There exists a sequence {Ii} whose associated parametrized measure is homogeneous and defined by

(Il, cp) = for any continuous, bounded cp.

I~I 10 cp(J(x)) dx,

121

3. Riemann-Lebesgue lemma

The proof of this fact is an immediate consequence of the homogenization theorem proved in the last section. For j fixed, let

be a Vitali covering of D,

where the subsets {aij

+ EijO}

are pairwise disjoint. Define

and conclude by the homogenization theorem. In particular, if we take 0 = (O,l)N and f E U(O) extended by periodicity to all of R N, the sequence Zj (x) = f (j x) determines the parametrized measure (7-3) cp; = I~I cp(f(x)) dx.

L

,v,

The reason why this is so is that due to the periodicity of Zj as

ZJ(.1:) = f(j(x- ai)),

so that

{a

x E ai

f

we can also write

1

+ -:- 0, .7

·N

+ 10}J

is a Vitali covering of O. Hence the conclusion follows .7 i=1 from Lemma 7.3. We can specialize even more. If f E LP(O) as before with 0 the unit cube in R N and extended by periodicity, for Zj = f (j x) we have i

Zj

~

r f(x) dx

In

in U(O),

taking cp as the identity in (7-3). This is what in many references is called the Riemann-Lebesgue lemma. In the very particular case in which we take f(x) = sinx or f(x) = cosx for x E (0, 27r), we obtain that the sequences {sin(jx)} and {cos(jx)} converge weakly in any U(O,27r) to the average of sin

122

Chapter 7. Analysis of Parametrized Measures

or cos in one period cell which is a. This in particular implies that the Fourier coeficients of functions in LP(a, 21l') tend to a. Another interesting consequence of this lemma is that for any domain n and any A E [a, 1] we can find a sequence of characteristic functions of subsets of n, Xj, such that Xj ~ A in U(n) for any p. Take any measurable set E c Lebesgue lemma to f = XE·

n such that lEI =

A Inl and apply the Riemann-

4. Two auxiliary lemmas

The two basic lemmas contained in this section are important in understanding the restrictions that a given family of probability measures should verify in order to ensure that it is the parametrized measure associated with some sequence of functions in some space LP(n). For y E Rm we introduce the set

My = {v, probability measure over R m

:

there exists

Z

E

10 z(x) dx = Inl y} .

Lg(n), v =

Dz(x),

8z (x) denotes the averaged parametrized measure associated with the constant sequence {z(x)} according to Theorem 7.1. Recall that £Y(n)

=

{z :n

-+

R m , measurable:

10 9 (lz(x)l) dx < oo}.

Note that for v E My

k",
My is a convex set of probability measures.

Proof Let Vi E My, i = 1,2 and A E (a, 1). There exist Zi E LY(n), Vi = DZi(x) and Zi(X) dx = Inl y. Let us take Den, a regular sub domain with IDI = A Inl. By Vitali's covering lemma, there exist two countable families of subsets, {aj + Ejn} and {b j + pjn}, pairwise disjoint and

In

D

= U(aj + Ejn) UN,

INI = a,

j

n\ D

=

U(b j j

+ pjn) UN', IN'I = a.

123

4. Two auxiliary lemmas We define

For any zp E [.g,

I~I L zp(z(x))dx = 1~12( [ff 10 zp(zddx+pf 10 ZP(Z2)dX] = A I~I

L

zp(Zl) dx

+ (1 -

A)

I~I

L

ZP(Z2) dx

= (AVl + (1 - A)V2' zp). This means that z E £9(12) (taking zp = g) and AVl

Lemma 7.5

+ (1- A)V2 = 8z (x)

E My .

•

For any continuous zp, the convexification of zp, zp**, is given by

zp**(y) = inf

{L=

zp(A) dV(A) :

vE My}.

Proof. On the one hand, by Jensen's inequality

zp**(y) for v E

:s:

lam CP**(A) dV(A) :s: l= cp(>.) dV(A),

My, so that cp**(y)

:s: inf

{l=

cp(A) dV(A) : v

E

My}.

On the other, if y = ~;~l AiYi with ~;~1 Ai = 1, Ai :;:, 0, choosing f2i C such that 1 f2i 1= 1121 Ai, and setting

n

zen)

=L ;=1

XOiYi,

12

124

Chapter 7. Analysis of Parametrized Measures

we have z(n) E

£9(0,)

and

'P**(Y) = inf {tAi'P(Yi): tAiYi = y, tAi = 1,Ai 2':

= inf { = inf { 2': inf

I~I

L

'P(z(n)) dx : z(n) = t

r 'P(A) d8

JR

m

{lm

z (n) (x) (A)

'P(A) dV(A) : v

E

: z(n) =

XOiYi,

:t,~l

o}

10,1 Ai = 100 i l}

XOiYi,

10,1 Ai = 100i I}

My } .

•

5. The homogeneous case Given a family of probability measures v = {vx } xEO depending measurably on x E 0" when can we find a sequence of measurable functions {Zj} such that v is the associated parametrized measure according to Theorem 6.27 We can say that there is no real condition that v should satisfy except for a technical assumption ensuring that the sequence of functions is bounded in LP(0,). We treat first the case in which v does not depend on x, the homogeneous case, and based on this we extend the result to the nonhomogeneous case in the next section. Let again the function 9 be fixed. Theorem 7.6

Let v be a probability measure supported in R17\ such that

lm

g(IAI) dV(A) < 00.

There exists a sequence of functions {Zj} such that {g( IZj I)} is weakly convergent in Ll (0,) and the corresponding parametrized measure is v, homogeneous.

Except for technical details, the proof consists in finding the sequence {Zj} by using the Hahn-Banach theorem: the measure v is shown to belong to the weak * closure of a convex set of measures where Dirac masses are dense. All homogeneous characterizations of parametrized measures are based on this same idea. Proof. Let us set

and consider My as a subset of ([g)'. Let T be a continuous, linear functional on ([g)' under the weak * topology, so that there is a'lj; E [g such that

125

5. The homogeneous case

for J.l E ([g)'. Let us suppose that T is nonnegative over My,

1 n

for all

Z

E

¢(z(x)) dx 2' 0,

£9(0,),10,1 y = / z(x) dx .

.!n

(y) 2' 0 and due to Jensen's inequality

Lemma 7.5 says that

(y)

s /

JRtn

Thus v cannot be separated from My (Hahn-Banach theorem), and by Lemma 7.4, v E co(My) = My. Since [g is separable, there exists a sequence {zJ} such that (7-4)

for ¢ in a countable, dense subset, 5, of [g. We can assume that g(I'>"I) E 5, so that {g(lzjl)} is uniformly bounded in L1(0,). By density, we can obtain that (7-4) holds for any ¢ E [g. If we now apply the existence theorem, Theorem 6.2, to {Zj} and use the averaging procedure Theorem 7.1 (observe that condition (7-4) does not change under this operation), we can assume that the parametrized measure associated to {Zj} is homogeneous, J.l. We would like to show that J.l = v. This is straightforward since for all rp E Co(RrrI) we immediately get (rp, V) = (rp, /1) as a consequence of (7-4). This identifies 1/ = /i.. Finally, we want to show that {g(lz)I)} is weakly convergent in L1(0,). By Chacon's biting lemma. t.here exists a nonincreasing sequence of measurable sets, {0,d, such that

Keeping in mind (74),

.!nk g(lzjl) dx =

lim lim /

k-vyv J-'>OC

lim lim ( / g(lzjl) dx -

k-'>oo .1-'>00

.!n

/ .!n\n

g(lzjl) dX) k

= k---+cx) lim (09(1'>"1),1/) 100kl = 0, and this implies that at least for a subsequence {g(lzj I)} converges weakly in L1(0,) because the integrals of g(lzjl) are uniformly small on the exceptional sets 12 k . •

126

Chapter 7. Analysis of Parametrized Measures

6. Characterization of parametrized measures We now deal with the general, inhomogeneous case. The passage from the homogeneous case to the nonhomogeneous is done by "assembling" or patching the individual measures through the Vitali's covering technique. Although there is a considerable amount of technicality involved (especially when we place more restrictions on the sequences) the idea is simple and natural. Theorem 7.7 Let v = {VX}XEr! be a family of probability measures in Rm depending measurably on x E n. A necessary and sufficient condition to find a sequence offunctions {Zj} such that {g(lzj I)} is weakly convergent in Ll (0,) and the associated parametrized measure is v, is (7-5)

Proof The necessity is clear because of the representation in terms of the parametrized measure. Let us show the sufficiency. If we can find a sequence {Zj} such that

for all ~ E rand i.fJ E S, where rand S are dense, countable subsets of Ll(n) and Co(Rm) respectively, this fact identifies v = {vX}XEr! as the parametrized measure associated to {Zj}. Condition (7-5) implies that

for a.e. a E n. Let N be the complement of such a's so that INI = o. By Lemma 7.9 in the Appendix for p = 00, q = 1 and taking rk(a) = 11k for all a E n \ N, we have

r ~(x)45(x) dx =

ir!

lim L45(aki) k--+oo

i

1 . ~(x) aki+E"r!

dx

(7-6)

for all ~ E L1(n), i.fJ E S where

E n \ N and the union is pairwise disjoint. For fixed aki and by Theorem 7.6 for the homogeneous case, we can find a sequence {z}i} with vak; as its parametrized measure. We define then

aki

127

6. Characterization of parametrized measures

where j = j(k, i) is chosen in the following way. Notice that this sequence is indexed by k rather than by j. Write r x S = Uk Dk, with Dk finite and Dk C D k+1' For k, i fixed, choose j so that

for (~,
= lim k-Hx)

=

L1 i

aki+EkiO

in ~(x)'i3(x)

~(x) dx 'i3(aki)

dx.

(7-8)

For notational convenience we now change Zk to Zj. Finally, the fact that {g(lzj I)} is weakly convergent in Ll (0) can be established exactly as in the homogeneous case. By Chacon's biting lemma, there is a nonincreasing sequence of measurable sets, {Od, such that

Because of (7-8) for

~

== 1,

lim lim ( g(lzj I) dx

k--->oo J--->OO

ink

= lim lim ( k--->oo J--->DO

inr g(lzj I) dx -

( g(lzj I) dX) in\n k

= lim ( (g(IAI), V x ) dx = 0, k--->DO

ink

because the function (g(IAI), v x ) is an Ll(O)-function (again due to (7-8)) . • The particular examples we are interested in are g(t) = t P for p > 1 and +00 for t 2 R which corresponds to the case p = 00. We close this chapter with one interesting example. Theorem 7.7 says that any family of probability measures can be generated by an appropriate sequence of functions. Let us try to construct explicitly a generating sequence for the family of probability measures

g(t) =

Vx

= (1 - x)8 1 + XLI,

X

E (0,1).

128

Chapter 7. Analysis of Parametrized Measures

For continuous cp, we would like to find a sequence

lb

cp(fj(x)) dx

->

lb l lb lb

such that

cp(>.) d((l - x)81 + xL 1 )(>') dx

[(1 - x)cp(l)

=

h

= cp(l)

+ xcp( -1)]

(1- x) dx + cp(-l)

dx

lb

xdx.

Let us assume that h takes on the values 1 and -1 in sets Aj and Ej = (0,1) \Aj respectively, such that IAj n [a, b]1 is a Riemann sum for the integral of (1 - x) in [a,b] and the same for IEj n [a,b]l. For instance, if

and we take

f·J = XA-

and since aj

-> a

J

lb

- XB

and bj

->

cp(h)dx

=

J'

then if a· J

= (aj) J

and b. J

= (bj) J

b,

cp(l) IAj n [a,b]1 + cp(-l) IEj n [a,b]1

->cp(l)

lb

(l-X)dx+CP(-l)

lb

xdx.

7. Appendix

°

1. For a given point x E R m , a sequence of sets {Ei} shrinks suitably to x if there is a > such that each Ei C E(x, Ti), a ball centered at x and radius Ti > 0, and

°

where Ti -> as i -> 00. A family of open subsets {A>J~EA is called a Vitali covering of n c Rm if for every x E n there exists a sequence {Ai} of subsets of the given family that shrink suitably to x.

7. Appendix

129

Theorem 7.8 Let A = {AAhEA be a Vitali covering oUt There is a sequence Ai E A such that

and the subsets AAi are pairwise disjoints.

The situation to which we apply the above covering theorem is the following. Let 0 be an open, bounded subset and B a ball containing O. The family of subsets Ak = { a + EO : a E

0, E< ~,a + d1" CO}

is a Vitali covering of O. Indeed, for any a E la + EOI _ ~ la+EBI - IBI

n we take a

_a - ,

for all

=

II~\ > 0 and

E.

Therefore by Theorem 7.8 0=

U(ajk + Ejk O ) UNk'

INkl = 0,

j

and the {ajk

+ EjkO}

are pairwise disjoint.

2. The following is a useful, technical lemma.

Lemma 7.9 Let 0 C RN be an open, bounded set with 1801 = 0 and NCO, a subset of measure o. For rk : 0 \ N ---+ R+ and {!J} c U(O), there exists a set of points {aki} C 0 \ N and positive numbers {Ekd, Eki :::; rk(aki) such that

{ aki

+ Ekin}

are pairwise disjoint for each k,

n = U{aki + Ekin} U N k ,

In ~(x)!J(x)

dx

= }~~~ ~ !J(aki) lki+€kirl.

for every j and every ~ E Lq(O), ~

+

! = 1.

~(x) dx

130

Chapter 7. Analysis of Parametrized Measures

Proof Let D c 0 be the intersection of the sets of Lebesgue points of the fj's and set A = 0 \ N. For each k the Lebesgue differentiation theorem implies that the family

-{ -.

11

a+EO.aEA,ESrk(a)'-1 01

Fk-

E

a+EO

Ifj(x)-fj(a)1 p dx<

1

k'

1 S j S k, and a + d'1 cO} covers A in the sense of Vitali. Hence we may write IN~I

= 0,

or

For any

110 ~(x)

~ E

Lq(O) and for fixed j, k 2: j

fj(x) dx -

=

IL 1 i

~ fj(aki) lki+ ki O~(x) dxl E

aki+t::ki r1

(iJ(x) - iJ(aki))

::; ~ (lki+ ki o I!j(x) c

~(x) dxl

!j(akiW

dX) lip (lki+ O1~(xW dX) l/q E

<::

~ ~ IEkiOI1/p (lki+£k,O 1~(xW dX)

<;

~ (~I"inl) '/p (~L.,," l«x)I' dX) 'I,

ki

l/q

= ~ 1011/p 11~IILq(O) . We have used Holder's inequality for integrals and for series.

•

8. Bibliographical remarks Homogenization and localization as have been presented here were introduced in [186] in the context of gradient parametrized measures to be studied in the next chapter. See also [278]. These turn out to be the basic tools that need to be redone for any new situation. The Riemann-Lebesgue lemma is a classical fact. It can also be found in [98] and textbooks dealing with Fourier Analysis.

8. Bibliographical remarks

131

The two auxiliary lemmas in Section 4 are well-known facts from elementary convex analysis. The rest of the chapter follows along the lines of [186] and [191]. The main facts shown in this chapter, under no differential constraint, are essentially contained as well in [93] and [302]. Concerning the technical facts in the Appendix, we refer to [283] for an accesible account on covering lemmas related to differentiation of measures. A complete treatment of Vitali's covering lemma as presented here can be found in [285]. There is a whole collection of covering lemmas that are extremely useful in Fourier Analysis. [291] is an standard reference for this topic. The last result is designed especially to be used in this book and its full proof has been included. The classical theorem of Radon-Nykodim and the Lebesgue differentiation theorem can al"o be found in [283].

Chapter 8 Analysis of Gradient Parametrized Measures

1. Introduction

In variational principles we are especially interested in integrands depending on gradients. For this reason we would like to study weak convergence associated to sequences of gradients. Parametrized measures associated to sequences of gradients are called gradient parametrized measures. In particular, we would like to prove a characterization for parametrized measures coming from a sequence of gradients. The paradigm in the calculus of variations is the functional

J(U) =

L

cp(x, u(x), \7u(x)) dx,

(8-1)

where 0 and cp are given, and the functions U are assumed to belong to some Sobolev class of continuous functions with weak derivatives. Let us assume we have a sequence offunctions in W1,P(0), {Uj}, converging weakly in W1,P(0) to some U E W1,P(0). We know that the functions themselves converge strongly in LP(O) to u, so that if we consider the parametrized measure associated to {(Uj, \7Uj)} then by Proposition 6.13, J.Lx =

8u (x)

Q9 Vx,

a.e. x

E

0,

where v = {Vx } xEO is the parametrized measure corresponding to the sequence of gradients. We need to fix our attention on the sequence of gradients and, in particular, we may consider without loss of generality as far as understanding weak lower semicontinuity is concerned, that the integrand cp in (8-1) only depends upon the gradient variable, so that

J(U)

=

L

cp(\7u(x)) dx.

(8-2)

In order to obtain a characterization of parametrized measures associated to sequences of gradients there is a fundamental difference between the scalar P. Pedregal, Parametrized Measures and Variational Principles © Birkhäuser Verlag 1997

134

Chapter 8. Analysis of Gradient Parametrized Measures

and the vector cases which makes them drastically distinct. If we deal with functions the scalar case corresponds to m = 1 whereas the vector case corresponds to m > 1. For such functions u, \lu E MmxN. If m = 1, \lu is in fact a vector and for any two given vectors Yl, Y2 E M 1xN we can always find a continuous function u such that \lu takes on only the two prescribed values Yl, Y2. This fact leads to the conclusion, after some manipulations, that if the functional I in (8-2) is weak lower semicontinuous, the integrand cp should be convex in the usual sense. In the case m > 1 what we have just asserted is not true: if you want to find a continuous u whose gradient \lu takes on only two values Y1 , Y2 E MmxN then Y1 - Y2 must be a rank-one matrix. There is no way to find such a u if Y1 - Y2 does not have rank-one. This is a fact which can be proven quite simply. Take any component of u, Ui, and suppose that \lUi takes on two values Yl, Y2 accross a plane interface with normal vector n. Let Xo be any point in this plane interface. Ui(X) = Yl . X + Cl if x . n > 0 for constants Yl, Cl E R. Similarly Ui(X) = Y2 . X + C2 if x . n < 0 for constants Y2, C2. The continuity of U at Xo implies YIXO+Cl = Y2XO+C2 or (Yl -Y2) 'Xo = C2 -Cl. Since this relation should hold for all Xo in the interface we have that (Yl - Y2) = ain, for some constant ai E R. Doing this for all the components of U we see that a necessary condition to find such u is Y1 - Y2 = a ® n for some vector a E Rm and n E RN. As we indicated in Chaper 1 this rank-one condition between Y1 and Y2 is also sufficient to find a continuous deformation u whose gradient takes on the two values Y1 and Y2 . This impossibility has the profound implication that the convexity condition can only be recovered along rank-one directions (8-3) provided that Y 1 - Y 2 is a rank-one matrix and the functional I is weak lower semicontinuous. See Chapter 1, Section 4 for more details. Plain convexity is no longer a necessary condition for the weak lower semicontinuity of I in the vector case. New types of convexity need to be introduced. In connection with the vector case, there are three new, relevant types of convexity: polyconvexity, quasiconvexity and rank-one convexity. The latter is given in (8-3). Quasiconvexity is the necessary and sufficient condition to have weak lower semicontinuity. cp is said to be quasiconvex if

cp(Y) ::;

1~11n cp(Y + \lu(x)) dx

(8-4)

for any matrix Y E MmxN and any smooth u with compact support in O. If cp is quasiconvex, of all deformations having affine boundary values given by Y, cp prefers precisely the homogeneous, affine deformation determined by Y.

2. Homogenization, localization and the Riemann-Lebesgue lemma

135

Polyconvexity provides :mfficient conditions for quasiconvexity since (8-4), which is hard to understand in practice. It turns out that parametrized measures generated by gradients are characterized by Jensen's inequality for all quasi convex functions (more precise statements arc needed concerning growth restrictions). The technique to show this result is essentially the same used in the previous chapter for the general case. As a matter of fact, we will try along this chapter to reproduce the same tools for the gradient case, point out where they fail and how difficulties can be overcome. Quasiconvexity will be closely examined along the way and some of its basic properties established. Throughout this chapter the function g will be taken to be g(t) = t P for 1 ::; p < 'Xl or g(t) = +Xl for t 2: R, some positive R, so that we will concentrate on V'(O)-bounds for sequences of gradients. In these cases we put £P and (£P)' for the appropriate spaces of functions to be considered and their duals: £P

= {cp : M mxN

--+

R, continuous : lim

cp(A)

IAI--->oo 1 + IAI P

eXists}.

We will also keep in mind the nonseparable spaces

XP =

{cp : M7nxN

--+

R, continuous:

Icp(A)1 ::; C(l + IAI P ), C

E

R} .

£P is a close subspace of XP under the same norm

cp(.) II Ilcpll = II -~p . 1 + 1·1 L=(MmXN) The dual space (XP)' can indeed be identified with the set of finite, Radon measures over MTnXN, IL, such that 1·11' belongs to L1(dfL). For the sake of brevity in sorne of our statements, we agree in calling a family of probability measures v = {VX }xEn supported in MmxN, a W1,p_ parametrized measure if it can be generated by the gradients of a bounded sequence of functions in Wi.P(O). vVe include here the case p = -Xl. This is the same terminology introduced in Chapter 1.

2. Homogenization, localization and the Riemann-Lebesgue lemma To begin with we examine the homogenization and localization procedures and the Riemann-Lebesgue lemma when working with gradients: the sequence Zj is now \111] for 11j E VV1,p(O). In this section we do not need to distinguish between the cases 1 ::; p < 00 and p = 00, or between the scalar and the vector cases.

136

Chapter 8. Analysis of Gradient Parametrized Measures

Let us look first at the homogenization process. Going back to the proof of Theorem 7.1, we realize that if we can find a sequence offunctions Wj E W1,P(O) such that 'V'Wj(X)

for x E aij

= 'V'Uj

(X

~jaij )

,

(8-5)

+ EijO, where once again by the Vitali's covering lemma

then the proof is exactly the same as that of Theorem 7.1. Condition (8-5) can be fulfilled by simply putting

for x E aij + EijO. It may not be true though that Wj so defined belongs to W1,P(O). Indeed, we know that Wj should be much more regular than simply belonging to LP(O). For p sufficiently large, Wj ought to be even continuous, and this might not be the case if we are not more careful about our definition of Wj. The only known way of having the continuity property for Wj is to enforce affine boundary values for Uj. Let Y E M mxN and let Uy be the affine function uy(x) = Yx. If Uj E W1,P(O) and Uj - Uy E WJ'P(O) the function

is well defined as a function in W1,P(O) since it is continuous (easy to check), Wj - Uy E WJ'P(O) and its gradient satisfies (8-5). Hence we have proved the following theorem. Theorem 8.1 Let {Uj} be a bounded sequence of functions in W1,P(O) with affine boundary values given by Uy. Let v = {vx } xEn be the parametrized measure associated with {'V'Uj}. There exists a sequence {Wj}, bounded in W1,P(O) with the same boundary values, such that the corresponding parametrized measure is V, homogeneous, and given by

Since the Riemann-Lebesgue lemma is a homogenization fact, its proof for the gradient case is based in the same idea. We need to have the precaution enforcing affine boundary values.

2. Homogenization, localization and the Riemann-Lebesgue lemma

137

Let 0 and D be two domains in RN (open, bounded, regular subsets) and U E W1,P(D), U - Uy E WJ,P(D). There exists a sequence {Wj} bounded in W1,P(0), Wj - Uy E WJ'P(O), such that the parametrized measure associated with the sequence {VWj} is homogeneous and given by

Lemma 8.2

(v, cp) =

1~ll cp(Vu(x)) dx,

for any cp E XP. The proof is again a basic application of the Vitali's covering lemma. For j fixed, let c· t) tJ { a t) +f··D:


j

be a Vitali covering of n,

where the subsets {aij .( ) _ W) X -

{

+ fij D}

are pairwise disjoint. Define

X - aij ) U ( -f-.-. t) uy(x),

Eij

+ uy(aij),

x

E

aij

+ EijO,

otherwise.

This lemma will prove to be important in understanding later on the concept of quasiconvexity from the point of view of parametrized measures. The other versions of this lemma previously discussed have also a counterpart with periodic gradients. They are left to the reader. We now turn to the localization property. Once more the proof is analogous to the general case provided we define properly the new sequence of functions giving rise to the new parametrized measure. In this case we should have Vuj,p(X) = VUj(a + px), p > O. As in Theorem 7.2, the sequence {Vuj,p} is uniformly bounded in LP(O) in j and p for a.e. a E n, and if v = {vx } xEn is the parametrized measure associated to {VUj}, then {Vuj,p} generates Va, homogeneous, for an appropriate subsequence. In order to have a bounded sequence of W1,P(0)-functions, {uj,p}' as p ---t 0, define

F(x) =

1

MmxN

Advx(A),

the linear function ua(x) = F(a)x, for a E 0, and its average over

IAI In ua(x) dx.

n, C a =

Take

uj,p(x) =

~(Uj(a + px) -

Ma,j,p),

x

E

0,

(8-6)

138

Chapter 8. Analysis of Gradient Parametrized Mea.sures

where the constant

Ma,j,p

is chosen so that

By Poincare's inequality, the sequence {uj,p} is truly bounded in W1,p(n) independently of j and p and for almost every a E n. We would also like to incorporate the affine boundary values for the new sequence defining the localized, homogeneous parametrized measure at almost every point in n. In order to do that we have the following lemma. Lemma 8.3 Let {Vj} be a bounded sequence in W1,P(0) such that the sequence {VVj} generates the parametrized measure 1/ = {I/ x } xE!1' Let

VU(x)

=

r

JMtnXN

Adl/x(A) E M mxN ,

U E W 1,P(0),

so that Vj ~ U in W1,P(0). There exists a new sequence {ud, bounded in W 1 ,P(0), such that {Vud generates the same parametrized measure v and Uk - u E W~'P(O) for all k. If for p < 00 {IVvjjP} is equiintegrable, so is

{IVUkjP}.

Proof Let {l7d be a sequence of cut-off functions with the properties: i) 17k = lover 00; ii) 17k == 0 in Ok = {x EO: dist(x, on) ~ 11k}; iii) IV1')kl <:::: Clk, for some constant C. We consider the functions

Clearly

Wjk -

U

E W~,p(n)

for all j, k. Moreover

Since Ilvj - ull LP(!1) converges to 0, a subsequence j (k) can be chosen so that the third term in the above sum tends to 0 and it is therefore weakly convergent in L1 (0). The first term is also weakly convergent. The middle term in the sum is also bounded or weakly convergent in U (0). Let {ud be the sequence {Wj(k)d, so that {IVUkjP} is either bounded or weakly convergent in L1(0), Uk - U E W~,p(n). The fact

implies by Lemma 6.3 that the parametrized measure associated to {VUk} is again v. •

139

3. The scalar case

If we apply the lemma to the subsequence of {uj,p} given in (8-6) we obtain our version of the localization property for gradients. Theorem 8.4 Let {Uj} be a bounded sequence in W1,p(n) and v = {vXLEn the parametrized measure associated to {VUj}. Let

and ua(x) = F(a)x, a E n. For a.e. a E n, there exists a sequence {wj} bounded in W1,p(n) such that wj - Ua E W~,p(n), for all j and the parametrized measure associated to {Vwj} is Va, homogeneous. 3. The scalar case Let us not make any distinction between the scalar and the vector case for the moment and try to reproduce the lemmas in Section 4 of the previous chapter in the special case when we are working with gradients. In this situation gradients are matrices in MmxN. We define the analogue of My by My = {v, probability measures over M mxN : v = U - Uy E W~,p(n)}

D\lu(x) , U E

W1,p(n),

c (t7)' ,

where Y E MmxN, Uy is the linear transformation Yx and Vx represents the averaged parametrized measure for Vx as discussed in Section 2. The reason why we impose a boundary condition on U rather than an average condition should be clear after the discussion of the averaging procedure.

Lemma 8.5

My is convex.

Proof Let Ui E W1,p(n), Ui - Uy E W~,p(n), i = 1,2 and let Vi be the average of the gradients of Ui as in the definition of My. Let A E (0,1). Take a regular, open subdomain Den such that IDI = ..\ Inl. There exist two countable families of subsets of D and n \ D of the type

{ai

+ fin: ai

E D,fi

> O,ai + fin c D}

{b i + Pin: bi En \ D, Pi > 0, bi + Pin c n \ D} such that

n \ D = U(bi + Pin) UN', IN'I = 0.

140 Define

Chapter 8. Analysis of Gradient Parametrized Measures U

by

+ uy(ai), x E ai + fin, Pi U2 (x Pi bi ) + uy(bi ), x E bi + Pin,

fi {

u(x) =

UI (x;i ai )

uy(x),

otherwise.

X -

Vu(x) =

{

a.)

VUI ( ~,

VU2

(x - b;) Pi '

As in the proof of the analogous lemma in the previous chapter, for 'P E EP we get

In 'P(Vu(x))dx L [ff In 'P(VuI)dx+pf In 'P(VU2)dX] • = ,\ In dx + (1 -,\) In 'P(VU2) dx =

'P(VUI)

= In particular U E WI,p(n)

+ (1 - '\)V2' 'Pl. and '\VI + (1 - '\)V2 E My. ('\Vl

•

In trying to prove the next lemma we see the basic difference between the scalar case and the vector case. Let us take first m = 1. Lemma 8.6

Let m

=

1 and let 'P be any continuous function. Then

'P** (Y) = inf

{r

JMIXN

'P(A) dv(A) : v E My} .

Proof Let us assume that we have Y = tYI + (1-t)Y2 where Y, Y I , Y2 E MIxN. They are in fact vectors. Consider the function

Xt is the characteristic function of the interval (0, t) in (0,1) extended by periodicity to all of R. We notice that VU(x)

=

Y2 + Xt((Y1 -

_ {Y1 , - Y2 ,

Y2) .

x)(Y1

-

(YI - Y2 ) • x - ((Y1 (YI - Y2 ) . x - ((Y1

Y2 ) -

Y2) . Xl < t, Y2) . Xl > t.

3. The scalar case

141

Define Uj(x) = (1/j)u(jx). We know that the associated parametrized measure IS

v = tOy!

+ (1 -

t)OY2

homogeneous. By Lemma 8.3 v E My (closure meant in the weak and

tcp(Y1 )

+ (1 -

t)cp(Y2) 2: inf {

r

JM1XN

* topology)

cp(A) dv(A) : v E My}.

Consider now the situation in which Y = 2:;=1 AiYi, a convex combination. Renaming and rearranging this sum we could write

+ (1 - t)y C2), = t(l)yP) + (1 - tCl))YP), = t(2)yP) + (1 - t(2))YP),

Y = tY(l) y(l) y(2)

for appropriate t, t(l), t(2). Applying the preceding arguments to y, yCl), y(2) we can find a sequence {vd whose gradients take on essentially the values yCl) and y(2) with relative frequency t and I-t respectively. For k and i = 1,2 fixed, let n~ stands for the set where \1vk = yCi) and replace n by n~ in the previous construction for a convex combination of just two matrices corresponding to the decomposition yCi) = tCi)ylCi) + (1- t Ci ))Y2Ci ), i = 1,2. We obtain sequences

{v}k} whose gradients take on the values, except on small sets Ejk c nL ylCi) and yY) with relative frequency large enough so that

t(i)

and 1- t( i) , respectively. Choose j = j (i, k)

IEjkl In~1

as k

-+ 00

uniformly in i

-+

0

= 1, 2 and define Uk(X) = {vjk(X), Vk(X),

if x E nL else.

These functions Uk are well defined as Wl,P(n)-functions since we are keeping track of the boundary values on an~. It is an interesting exercise to realize that if Vk is the average of the gradient of Uk, by the constructions carried out,

r

JM1XN

t(t(1)cp(YP))

+ (1 -

t(1))CP(Y2Cl )))

cp(A) dVk(A)

+ (1 -

-+

t)(t(2)cp(YP))

+ (1 -

t(2))cp(YP))),

as k -+ 00. Since each Vk E My, because the boundary condition is kept throughout the above process,

142

Chapter 8. Analysis of Gradient Parametrized Measures Generalizing this construction recursively for any finite number of vectors

Yi and numbers ti and using Jensen's inequality, we get '11** (Y) = inf {

?': inf

8 n

n

tiip(Yi) : ti > 0, ~ ti = 1,

{1

M,xN

?': inf{l

8 n

tiYi = Y

}

ip(A) dv(A) : v E My} ip**(A)dv(A): v E My}

M1XN

?': ip**(Y). Notice that for any U E W 1,P(f2) such that U - Uy E W~'P(f2) it is true that if v = 8\lu, by the divergence theorem

1

M'XN

A dv(A) =

I~I inrVu(x) dx = Y. H

The proof is finished.

•

With these two lemmas at hand one can reproduce almost exactly the proofs of Theorems 7.6 and 7.7, and obtain the corresponding characterization for the scalar case when m = 1. There is an important difference in the fact that we are working with gradients so that proofs change somewhat. Since we will redo these proofs for the vector case in subsequent sections, we refer the reader to those sections for the proofs.

Theorem 8.7 Let v = {v x } xEn be a family of probability measures supported in Ml x N. The necessary and sufficient conditions to find a bounded sequence of functions, {Uj}, in W 1,P(f2), such that for 1 < p < 00, {IVujn is weakly convergent in U(f2), and the parametrized measure associated to {VUj} is v, are Advx(A), for some U E W 1,P(f2), Vu(x) =

1

M,xN

and

1

inr where K

c Ml x N

M,xN

IAI P dVx(A) dx <

supp (V x )

c K,

00,

a.e. x E f2,

1 s:; p

p=

< 00,

00,

is a fixed compact set.

This theorem is not valid for p = 1. In this case, even though {Vj} is a bounded sequence in W 1,1(f2) it may not be true that Vj ~ v in W 1,1(f2) with v E W 1,1(f2) so that there is no way to know if the first moment 1M Advx(A) is the gradient of a Wl,1 (f2)-function. If we explicitly assume this as fact, the theorem is valid for p = 1 as well.

4. Quasiconvexity

143

4. Quasiconvexity

Let us now deal with the vector case m > 1 and take p = 00 for the rest of this section. In this case the set My is a set of probability measures compactly supported in M mxN since the gradients of functions in W1,OO(0) are bounded. We simply set My =

{I/,

probability measure: u -

Uy

E

1/

= 8vu (x),U E W1,OO(0),

W~,OO(O)}.

Assume we would like to redo the proof of Lemma 8.6 in this case. Given two matrices we need to find a continuous function u such that its gradient '\lu takes on two values Y 1, Y2 E MmxN. The continuity of u is necessary because if such u is to exist its gradient would be bounded and therefore u E W1,OO(0). This condition places an important restriction on Y 1 , Y 2 . In the Introduction and in Chapter 1 we have argued that a necessary and sufficient condition for finding such a Lipschitz function u is Y 1 - Y2 = a @ n for some vector a E Rm and n ERN. Indeed, if Y 1 - Y2 = a @ n then we can take

In this case

'\lu(x) =Y2 + Xt(n· x)a@ n

_{Yl'

-

Y2 ,

n·x-(n·x} t,

and we could redo Lemma 8.6. The condition Y1 - Y 2 = a@ n places a real and fundamental restriction so that in general the infimum inf { (

JMmxN

cp(A) dl/(A) : 1/ E My}

(8-7)

is no longer cp**(Y). For example, take m = N = 2 and let cp(A) = det A. Lemma 8.8

Let

u

E W1,OO(0) such that

10 det('\lu(x)) dx

u -

=

Uy

E W~,OO(O). Then

101 det Y.

Chapter 8. Analysis of Gradient Parametrized Measures

144

Proof Let r.p be a cgo(n) and consider the function v(x) claim that

In

= uy(x) + r.p(x). We

det(Vv) dx = 10,1 det Y.

In fact det(\7v) = det(Y + \7r.p) = det Y + det \7r.p - (adjyf . \7r.p. The third term on the right-hand side involves some partial derivative of some component of r.p multiplied by some component of Y. Since r.p E cO'(n), by the divergence theorem the integral over 0, of these terms vanish. Concerning the second term it is very easy to obtain

Again by the divergence theorem we have that the integral of this term also vanishes. The result follows approximating u - Uy by a sequence of smooth functions in the strong W1,OO(n)-topology (observe that det is continuous in the strong topology of W1,OO(n)). • For v E My there exists u E W1,OO(n), u - Uy E wJ,OO(n) and by the lemma

L2X2 det A dv(A) = I~I

In

det(\7u(x)) dx

=

det Y.

This means that in fact det(Y)

=

inf

{rJM2X2 det(A) dv(A) : v

E

My}

even though det is not convex as can be checked very easily. Given a continuous function r.p defined over matrices, the above infimum (8-7) is called the quasiconvexification of r.p at Y and is denoted by Qr.p(Y). Equivalently

Qr.p(Y) =

I~I inf

{In

r.p(\7u(x)) dx : u E W1,OO(n), u - Uy E

wJ,OO(n)} .

As we will show shortly this definition does not depend upon 0,: the infimum is the same for all regular domains. We say that a continuous function r.p is quasiconvex if it coincides with its quasiconvexification for every matrix Y, i.e.

r.p(Y) =

I~I inf

{In

r.p(\7u(x)) dx : u

E

W1,OO(O), u - Uy E WJ'OO(O)} ,

4. Quasiconvexity

145

for all Y. In particular, det is a quasiconvex function on 2 x 2 matrices. An equivalent way of declaring a function r.p quasiconvex is by requiring r.p(Y)

:s: I~I

L

r.p(Y

+ \7u(x)) dx,

for all U E WJ'oo (D). As pointed out, we could repeat the proof of Lemma 8.6 if we have the condition Y 1 - Y 2 = a 0 n and this condition is also necessary if at any time we would like to obtain a W1,OO-function in the appropriate domain whose gradient takes on the values Y 1 and Y 2 across a plane interface with normal n. To emphasize this idea, a function r.p is called rank-one convex if

whenever Y1 - Y2 = a 0 n is a rank-one matrix. It should be clear that if r.p has this rank-one convexity property, the value r.p(Y) would not be lowered by the processes described in the proof of Lemma 8.6. As a consequence, if r.p is quasiconvex then it also is rank-one convex. The converse is not true. Said differently, it is not the case in general that the infimum in (8-7) is obtained by the scheme described in the proof of Lemma 8.6. We will examine in detail this question in Chapter 9. We now prove four basic properties of quasiconvexity. The first lemma establishes that the quasiconvexification is well defined and does not depend on the particular regular domain D that we use in its definition. Lemma 8.9 Let D and D' be two open, bounded subsets oERN with laDI = laD'1 = 0, and let r.p be a function defined on MffixN. For any matrix Y, the

two infima

I~I inf

{In

r.p(\7u(x)) dx : u E W1,OO(D), u -

Uy

E

WJ,OO(D)} ,

and

I~'I inf {Lf r.p(\7v(x)) dx : v E W1,OO(D'), v -

Uy

are equal. Proof By Vitali's covering lemma, we can write

D' = U(ai

+ fiD) n N,

INI = 0,

E

wJ'OO(O')}

146

Chapter 8. Analysis of Gradient Parametrized Measures

where ai Ell', Ei > 0, ai + Eill C 11' for all i. Observe that 111'1 = For any given u E W 1 ,CXJ(Il) such that u - Uy E W~,CXJ(Il), consider

v(y) =

{Ei (Y ~i a i ) + uy(ai), U

Uy

(y),

It is easily verified that v E W 1 ,CXJ(Il'), v -

If we interchange the roles of

11

and

11'

Uy

Li Efllll·

Ei

if y E ai + ll , otherwise. E W~,CXJ(Il') and

we obtain the desired result.

•

The second property states that the quasiconvexification of a continuous function 'P is indeed quasiconvex as one would reasonably expect. Lemma 8.10 Let'P be an upper-semicontinuous function defined on MmxN. The quasiconvexification of'P is a quasiconvex function. Proof. We proceed in several steps. Step l. Let u be a piecewise affine function such that u for some fixed matrix Y. We can set

u(X) = ai

11 = Then

L

UIli'

+ FiX,

X E Ili' ai E R m

Uy

,

11;, pairwise disjoint open subdomains.

Q'PC'vu) dx =

L, 111;1 Q'P(l'i)

E W~,CXJ(Il)

4. Quasiconvexity

147

It is easy to ascertain by a patching procedure that the last sum of infima is actually equal to inf

{l

cp(vrw) dx : w E w1,OO(n), w - Uy E w - U E W~,OO(ni)' for all

Finally,

i

Qcp(vru)dx =inf

~ inf

{i

w~,OO(n),

i} .

cp(vrw)dx: w E W1,OO(n),w - Uy E

{L

w - U E W~,OO(ni)' for all

W~,OO(n),

i}

cp(vrw) dx : w E W1,OO(n),

w - Uy E W~,OO(n) }

=Qcp(Y). Therefore the inequality (8-8) holds for piecewise affine functions. Step 2. Qcp is upper-semicontinuous. Let Y be any matrix and lj a sequence converging to Y. Let U be any W1,OO(n) function such that U-Uy E W~,OO(n). Consider Uj(x) = u(x) + (ljY)x. Then Uj - uYj E W~,OO(n) and therefore

Qcp(lj) ::;

I~I

i

cp(vrUj) dx =

I~I

i

cp(vru + lj - Y) dx.

By the upper semi continuity of cp limsup J-->=

rcp(vruj)dx::; inrcp(vru)dx,

in

and

Since

U

is arbitrary

lim sup Qcp(lj) ::; Qcp(Y).

Step 3. Conclusion. Given any U E W1,OO(n) such that U - Uy E W~,OO(n) there exists, by Theorem 8.20 (see the Appendix, Section 8), a sequence of piecewise affine functions, {Uj}, such that

Uj

----t

Uj - Uy E W~,OO(n), U in W1,p(n), 1 ::; p <

00,

Ilvrujlluoo(n) ::; K Ilvrullvx>(n)'

148

Chapter 8. Analysis of Gradient Parametrized Measures

Using these estimates on 'VUj we obtain for some M

>0

uniformly in D. By Fatou's lemma and step 2

inr (M -

Qip('Vu)) dx :s:: li!llinf )-+00

inr (M -

Qip('VUj)) dx.

We conclude by step 1

r Qip('Vu)dx ~ lim sup inr Qip('VUj)dx

in

)-+00

~

Qip(Y),

•

so that Qip is quasiconvex.

The next fact refers to the continuity of quasiconvex and rank-one convex functions. It is enough to show that rank-one convex functions are continuous. Lemma 8.11

Every finite rank-one convex function is continuous.

Proof. Observe that given any two matrices A and B, we can find at most n = min(m, N) matrices Ai, i = 1, ... , n, such that Al = A, An = Band rank (Ai - Ai-I) :s:: 1 for i = 2, ... ,n. In fact, we can change one full column or row at a time depending on whether n = N or n = m. Let A be fixed and assume that Aj -+ A. It is clear that A{ -+ A for all i, and in particular A{ - ALI -+ 0 for all i. Notice that A~ = A and A{ = Aj for all j. Since ip restricted to the lines determined by A{ and ALI is convex, this restriction is continuous as is already well known. Thus

•

= cp(A).

The last property is a different description of the quasiconvexification of a function that is convenient in some regards. Lemma 8.12 Let cp : MmxN following identity holds

Qcp

-+

R* be an upper-semicontinuous function. The

= sup {7/J : 7/J :s:: cp,7/J

quasiconvex} .

5. Wl,P-quasiconvexity

149

Proof Let Qcp denote the sup in the right-hand side. Since by Lemma 8.10 Qcp is an admissible function for such supremum, Qcp ::; Qcp. Conversely, if'ljJ is any admissible function in the supremum, for any U E WJ'oo (0) 'ljJ(Y) ::;

I~I

L

'ljJ(Y

+ V'u) dx ::;

I~I

L

.

cp(Y + V'u) dx.

Taking the infimum in u, we get 'ljJ(Y) ::; Qcp(Y). The arbitrariness of 'ljJ leads ~~::;~. 5. W1'P-quasiconvexity

In general, for finite p we say that a function cp defined on matrices is W1,p_ quasiconvex if for every Y E MffiXN cp(Y) ::;

I~I

L

cp(V'u(x)) dx,

for every U E W1,P(O) such that U ~ Uy E W~'P(O) and any regular domain O. Similarly the W1,P-quasiconvexification of cp is defined via the infimum

QPcp(Y)

=

I~I inf

{L

cp(V'u(x)) dx : U E W1,P(O), U ~ Uy E

W~'P(O)} .

Clearly cp** ::; Qlcp ::; QPcp ::; QqCP ::; Qcp for 1 ::; p ::; q. Notice that Qcp is in reality Qoocp. The following fact relates the property of Wl,P-quasiconvexity to quasiconvexity. In this context quasiconvexity is W1,OO-quasiconvexity. Lemma 8.13

Let cp be a continuous function defined on matrices such that

c::; cp(A) ::; C(l + IAn, C> O,p cp is W1,p -quasiconvex if and only if cp is quasiconvex.

2 1.

(8-9)

Proof We have to show that if cp is quasiconvex, it is W1'P-quasiconvex. Let Y be any matrix and U E W1,P(O) such that U~Uy E W~'P(O). Let Uj E W1,00(O) be a sequence converging to U strongly in W1,P(O) and Uj ~ Uy E W~,oo(O). By (8-9)

and by Fatou's lemma

P Inr(C lV'ul ~ cp(V'u)) dx ::; liminf Inr(C lV'ujlP ~ cp(V'Uj)) dx. )-+00

This implies that limsup ( cp(V'uj)dx::; ( cp(V'u)dx, )-+00

In

and since cp is quasi convex

cp(Y) ::; so that cp is Wl,P-quasiconvex.

L

In

cp(V'u) dx,

•

Chapter 8. Analysis of Gradient Parametrized Measures

150

In particular, under the bounds (8-9) the Wl,P-quasiconvexification of a continuous function is itself Wl,P-quasiconvex (indeed quasiconvex according to Lemma 8.10) and continuous. Under no growth assumptions, there are still many open, delicate issues concerning W1,P-quasiconvexity. 6. The vector case: proof of necessity We would like to understand the conditions that a family of probability measures should satisfy when coming from a sequence of gradients uniformly bounded in LP(n). There are three constraints that turn out to be sufficient as well: a spatial compatibility condition, a fundamental property in terms of Jensen's inequality for each individual member and a technical hypothesis. These are i), ii) and iii) below, respectively. Theorem 8.14 Let {Uj} be a bounded sequence in W1,p(n), p > 1, and let 1/ = {1/:1J"EO be the parametrized measure associated to the sequence of gradients {V'Uj}. Then i) V'u(x) = IwnxN Adl/x(A), for a.e. x E 0. where U E W1,p(n); ii) IMmxN cp(A) dl/x(A) 2: cp(V'u(x)) for every cp E £P quasiconvex and bounded from below and a.e. x E 0.; iii) IwnxN IAI P dl/x(A) dx < 00.

Io

For p = 00 ii) and iii) should be changed to ii') IMmxN cp(A) dl/x(A) 2: 'P(V'u(x)) for a.e. x E 0. and any 'P bounded from below and quasiconvex; iii') supp I/x c K for a.e. x E 0. where K c MmxN is some fixed bounded set. Assertions i) and iii) are quite simple. In fact, i) is just the representation formula in terms of parametrized measures of the first moment of I/x . Concerning iii), observe that by Theorem 6.11

For p = 00, iii') is even easier to prove since the sequence {Uj} is uniformly bounded in W1,00(n) and therefore the corresponding parametrized measure must be compactly supported uniformly in x E n. In order to prove ii), let us take any x E 0. fixed and write for the moment 1/ = I/x, Y = V'u(x). By the localization principle, Theorem 8.4, for a.e. such x E 0., there exists a sequence, {Vj}, converging weakly in W1,p(n) to Uy and such that Vj - Uy E wJ,p(n) for all j. Moreover, the parametrized measure corresponding to {V'Vj} is 1/, homogeneous. Again the case p = 00 is especially easy to deal with. Assume that {Vj} is bounded in W1,OO(n). If 'P is quasiconvex (and thus continuous), since {V'Vj} is uniformly bounded in Loo(n),

6. The vector case: proof of necessity

151

the sequence (or some suitable subsequence) {'P(\7Vjn is weakly convergent in L 1 (n). Therefore

On the other hand, 'P being quasi convex,

for all j. Hence we obtain

which is ii'). The real problem with finite p lies in the lack of weak compactness The following remarkable lemma shows how to for the sequence {'P(\7Vj overcome this fundamental difficulty. We turn back to the case where {Vj} converges weakly to Uy in W1,p(n) for 1 < p < 00. We refer the reader to the Appendix, Section 8, for some of the notation to be used and some fundamental results about maximal operators to be utilized in the proof of the lemma.

n.

Let {Vj} be a bounded sequence in W1,p(n). There always Lemma 8.15 exists another sequence {Uj} of Lipschitz functions (Uj E W1,oo (n) for all j) such that {1\7uj is equiintegrable and the two sequences of gradients, {\7Uj} and {\7Vj}, have the same underlying W1,P-parametrized measure.

n

Proof. Step 1. Assume furthermore that Vj E CO"(RN) and replace n by RN. Consider the sequence {M*(vjn where M* is the maximal operator of a function and its gradient. By the remarks recalled in the Appendix, this sequence is bounded in LP(RN ). Let /-l = {/-lX}xERN be the corresponding parametrized measure (possibly for an appropriate subsequence). Consider the truncation operators Tk defined by

Chapter 8. Analysis of Gradient Parametrized Measures

152

We have used the monotone convergence theorem for the second limit. Notice that

is a L1(RN)-function. We can find a subsequence k(j) ~ that

00

as j ~

00

such

On the other hand, by the observation about these truncation operators made after the proof of Lemma 6.3, the parametrized measure associated to the sequence {Tk(j)M*(V'vj)} is also fl,. By Corollary 6.10, we conclude that

Let

Aj

= {M*(vj) > k(j)}.

Then IAjl ~ 0 because {M*(vj)} is bounded in LP(RN ) and k(j) ~ 00. By Lemma 8.21, there exist Lipschitz functions Uj such that Uj = Vj (and therefore V'Uj = V'Vj) outside of Aj and, moreover,

IV'Uj I :::; C(N)k(j),

for all j.

The fact that IAj I ~ 0 implies that the parametrized measure for both sequences is the same (Lemma 6.3). It follows easily (M*(vj) 2:: lV'vjl) that

Since the right-hand side is equiintegrable in Ll(RN) the conclusion of the lemma follows. Step 2. Approximation. We can assume that Vj ~ U in W1,p(n) for some U E W1,p(n). Moreover, by Lemma 8.3, we can assume that Vj - U E W5,p(n). Let Wj = Vj -u extended by 0 to all of R N. By density, we can find Zj E CD (RN) such that

IIZj - Wj Ilwl,P(RN) ~ 0,

j ~

00.

Apply Step 1 to {Zj} and find a sequence of Lipschitz functions, {Uj}, such that {1V'uj is equiintegrable in Ll(RN) and I{V' Zj i- V'Uj} I ~ O. Therefore, again by Lemma 6.3, the parametrized measure for the sequences (considered now restricted to n) {V'Uj} , {V'Zj} and {V'Wj} is the same. Take Uj = ujlo + u. The sequence {Uj} verifies the conclusion of the theorem (see Step 2 of the proof of Theorem 8.16). •

n

153

7. The vector case: proof of sufficiency

For proving ii) in Theorem 8.14 in the case p finite, take a bounded sequence in W1,p(n), {Uj}, generating v = Vx for fixed x E n. By the lemma just proved, and using Lemma 8.3, we may assume that Uj - Uy E W~,p(n) where Y = V'u(x) and {1V'uj is equiintegrable. In this case, if i.p E £P is quasiconvex, it is in particular W1'P-quasiconvex and

n

Inl i.p(Y)

::; lim )-->00

1 n

i.p('\lUj) dx =

r

iM"'XN

i.p(A) dv(A).

This ends the proof of Theorem 8.14. Theorem 8.14 is valid for p = 1 if we assume explicitly Uj ~ U in W1,1(n).There are however a few steps in the proof that need to be fixed. We do not pursue this direction here.

7. The vector case: proof of sufficiency

This section is devoted to the proof of the result concerning the sufficiency part of Theorem 8.14. In this form it is also valid for p = 1. Theorem 8.16 Let v = {v x } xEIl be a family of probability measures supported on MmxN such that i) V'u(x) = ii)

IMmxN

Advx(A) for some U E W1,p(n);

i.p(A) dVx(A) ~ i.p(V'u(x)) for a.e. x E bounded from below and quasiconvex;

IMmxN

n

and for any

i.p E

£P

In

IM",xN IAI P dVx(A) dx < 00. iii) There exist functions Uj E W1,P(O) such that {1V'ujIP} is weakly convergent in Ll(n) and the parametrized measure associated to {V'Uj} is v.

The idea behind the proof is natural. We first take care of the homogeneous case when we do not have any spatial dependence on v. Property ii) says that if Jensen's inequality holds for all suitable quasiconvex functions then v can be generated by a sequence of gradients. The inhomogeneity of v is taken care of by an assembling procedure: we patch together many different individual V X ' We begin by treating first the homogeneous case. Proposition 8.17

Suppose that J1, E (£P)' is a probability measure for which

(8-11) whenever

i.p E

£P. Then J1, is a homogeneous Wl,P-parametrized measure.

154

Chapter 8. Analysis of Gradient Parametrized Measures

Proof We use the Hahn-Banach theorem. Let T be a linear functional on (£P)' in the weak * topology such that T :::: 0 on My, a convex set by Lemma 8.5 (the proof of this lemma is also valid for the vector case exactly as it stands). There exists 'ljJ E £P such that

o ~ (T, v) = ('ljJ, v) = For

v =

8'1lu,

U E

W1,P(O),

r

JMfflXN

u - Uy E

o~

'ljJ(A) dv(A),

v E My.

WJ'P(O),

In 'ljJ('\lu) dx.

(8-12)

Therefore, Q'ljJ(Y) :::: O. Thus by (8-11),

o ~ Q'ljJ(Y) ~

r

JMfflXN

'ljJ(A) dll(A)

= (T,Il).

Therefore, 11 E co(My) = My where closure is meant in the weak * sense. Since £P is separable, bounded sets in (£P)' endowed with the weak * topology are metrizable, and convergence can be characterized by sequences. Hence in a bounded neighborhood of 11 there exists a sequence {uk} C W1,P(O), Uk -Uy E WJ'P(O) such that

r

JMfflXN

'ljJ(A) dll(A)

= lim

k-+oo

r 'ljJ('\lu k) dx

Jo

for any 'ljJ E £P.

(8-13)

Let v be the W1,P-parametrized measure associated to {'\lu k }. By the averaging procedure we may assume v to be homogeneous (notice that (8-13) does not change in this process). Clearly 11 = v, since as a consequence of (8-13), ('ljJ,Il)

= ('ljJ,v)

• Theorem 8.18 A probability measure 11 in (£P)' is a homogeneous W1,p_ parametrized measure if (8-14)

for every 'P E £P which is quasiconvex and bounded from below. Proof Assume 'ljJ E £P, and set 'ljJn

= max('ljJ, (};n) = 'ljJX{'l/J~On} + (};nX{'l/J
where (};n ~ -00. 'ljJn ~ 'ljJ monotonically and 'ljJn E £P. The following lemma asserts that Q'ljJn does belong to £P as well.

155

7. The vector case: proof of sufficiency Lemma 8.19

If'lj; E £P and'lj; 2': G tben Q'Ij; E £P (and Q'Ij; 2': G).

Proof Assume first that G = O. Let .

'Ij;(A)

hm [A[P = a 2': O. IAI-->oo 1 + If a

= 0 then trivially lim Q'Ij;(A) IAI-->oo 1 + [A[P

as well. Let a

=0

> 0, and 0 < E < a. There exists ME such that for [A[ 2': ME 'Ij;(A) 2': (a - E) [A[P

+ (a - E),

a - E> 0

On the other hand 'Ij;(A) 2': -GE, OE > 0, if [A[ :::; ME' Altogether we have for any A Since the right-hand side is a convex function, we conclude by Lemma 8.12,

Q'Ij;(A) 2': (a - E) ([A[P - Mf) - GE , and taking limits for [A[

~ 00

we get

. .

Q'Ij;(A)

hmmf [A[P 2': a-E. IAI-->oo 1 + The arbitrariness of

E

> 0 and the fact that Q'Ij; :::; 'Ij; enables us to write

. . Q7jJ(A) . 7jJ(A) a :::; hm mf [A[P:::; hm [A[P IAI-->oo 1 + IAI-->oo 1 + This implies the conclusion of the lemma. If 0 is not 0, we apply the preceding arguments to

=

a.

;j; = 'Ij; - O.

•

We go back to the proof of Theorem 8.18. By hypothesis, since Q'Ij;n E £P is quasi convex

But by monotone convergence

i"'XN 'lj;n(A) dv(A) ~ i"'XN 'Ij;(A) dv(A). We now use Proposition 8.17 to conclude.

•

156

Chapter 8. Analysis of Gradient Parametrized Measures

Proof of theorem 8.16. Step 1. Assume the function U E W 1 ,p(n) in i) and ii) is O. It is sufficient to find a sequence of W 1 ,P(n)-functions with the property

r~(x)
lim

k->oo in

for all ~ E rand