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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
')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
1. If
1. If
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:
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 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 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
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 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
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{lfjloo 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
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 00
in
~(x)\)dx
i R",
(6-5)
for ~ and
'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=
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~lfjmlCXlm~n
1
r:"::{lfi m l}\!1 n
::; lim sup { r->oo
m~n J{r:"::lfiml 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
-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
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)
()
~
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
.) 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 (~,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
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/Joo 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)oo
= as desired.
2:1 i
a~+f~n
in ~(x)"ip(x)
dx,
~(x)dx "ip(af)
157
8. Appendix Step 2. Given a family v
= {vx}xEn
satisfying i), ii) and iii), consider
It is elementary to check that v is in the situation of Step 1 (we can always normalize the quasiconvex functions entering in ii) by requiring cp(O) = 0). Therefore there exists a sequence {Vj} generating v as a Wl,P-parametrized measure and such that {1V'vj jP} is equiintegrable. We claim that Uj = Vj + U generates v. If this is so we have completed the proof of the theorem. In order to see that claim, let 'ljJ(x, A) be a Caratheodory function and let {;(x, A) = 'ljJ(x, A + V'u(x)), itself a Caratheodory function. Hence lim )->00
lnr 'ljJ(x, V'Uj) dx =
lim )->00
lnr {;(x, V'Vj) dx
rr = r r ln = r r ln =
ln lM'nxN lM'nxN lMmxN
{;(x, A) dVx(A) dx {;(x, A - V'u(x)) dVx(A) dx 'ljJ(x, A) dVx(A) dx.
The arbitrariness of 'ljJ implies that the parametrized measure associated to {V'Uj} is v. • Theorem 8.16 remains true for the case p = 00 changing conditions ii) and iii) to ii') and iii') as in the previous section. The proof of this involves some further technicalities although the tools are the same as in Theorem 8.16. Since the case p = 00 is not relevant in Chapters 3 and 4, we do not include the proof here.
8. Appendix 1. We recall an approximation result by piecewise affine functions which is very useful in many different settings. A function U E W1,00(O) is called piecewise affine if 0 can be decomposed in a finite union = Ui i and V'u is constant on each Oi'
n
n
Theorem 8.20 Let 0 be a bounded domain with Lipschitz boundary and U E W~,OO(O). There exist functions Uj E W~,OO(O), piecewise affine, such that Uj
--+
U in W1,P(O), 1 :::; p
O
is the maximal function of
C(RN) and
f.
+ M(IV'v(x)I),
r
Br } Br(x)
If(z)1 dz
It is well known that if v E Cgo(RN), M*v E
and, in particular, for any A > 0,
I{M*v ::::- A}I
-s: C(N,p)r p Ilvll~!1.P(RN)'
1 O. Set HA = {M*v < A}. Then
Iv(x) - v(Y)1 < C(N);, HA Ix _ YI ,x, Y E , where C(N) depends only on N. It is also interesting to remember that any Lipschitz function defined on a subset of RN may be extended to all of RN without increasing its Lipschitz constant.
9. Bibliographical remarks The basic ingredients of homogenization and localization in the format developed here are contained in [191]. The basic construction on the proof of Lemma 8.6 has been known for many years. All the material related to quasiconvexity can also be studied in many different sources. Indeed, a fairly large number of works in the last two decades have been directed towards the understanding of the quasiconvexity condition and the property of weak lower semicontinuity for variational integrals. Once again we do not try to exhaust the bibliography on this subject. Some of these references are [2], [3], [4], [29], [37], [61], [95], [96], [97], [98], [151]' [166], [193], [216], [220], [225], [233], [238], [269], [289], [309]. The rank-one convexity condition has also been investigated extensively: [32], [96], [256]' [257], [295]' [307], [319]. The Wl,P-quasiconvexity condition has been studied in great detail in [38]. We refer to this paper for examples and further discussion.
9. Bibliographical remarks
159
The proof of the necessity part for Theorem 8.15 is based on ideas and technical results introduced in [145]. It is somehow a shorter, more direct version than the one contained in [4]. An alternative approach based on extra integrability for minimizing sequences was developed in [226]. The sufficiency part, Theorem 8.16, has been taken from [191]. Recently, another method has been proposed in [200] based on the Hodge decomposition. The facts in the Appendix are well known. The approximation by piecewise affine functions can be found in [118]. [291] is a standard reference for basic properties of maximal operators. See also [135].
Chapter 9 Quasiconvexity and Rank-one Convexity
1. Introduction
The motivation for this chapter is two-fold. On the one hand, since Jensen's inequality has played a prominent role in our approach to weak lower semi continuity, our analysis would be somehow incomplete without any reference to this inequality with respect to rank-one convex functions. Because quasiconvexity implies rank-one convexity, probability measures satisfying Jensen's inequality with respect to the class of rank-one convex functions are indeed examples of gradient parametrized measures. It turns out that this family of probability measures can be understood. at least conceptually, in a nice constructive way. They are called laminates to emphasize its layering structure. As a matter of fact, laminates are almost the only way to produce explicitly examples of gradient parametrized measures. It is true that the Riemann-Lebesgue lemma allows one to consider gradient parametrized measures associated with periodic gradients. The problem is that we do not know how to decide whether they are laminates or not. The importance of laminates in the description of some equilibrium states for crystals has been stressed in Chapter 5. They are also important in the theory of composite materials ane! homogenization. The second goal of this chapter is to show that rank-one convexity does not imply quasi convexity in generaL There is a duality between gradient parametrized measures and quasi convexity, and laminates and rank-one convexity. Jensen's inequality is the link. In this sense, the problem of deciding if rankone convexity implies quasiconvexity is equivalent to deciding if every gradient parametrized measure is a laminate. The two problems arc equally difficult at first sight. We will show an explicit example of a gradient parametrized measure (a microstructure) constructed through the Riemann-Lebesgue lcmma from a periodic gradient which is not a laminate. At the same time, we will find an explicit example of a rank-one convex function that is not quasiconvex. Both examples are intimately connected. They are valid only if the dimension of the target space for deformations is three or more. For dimension two the problem is still open. We will consider a direct extension of the counterexample to P. Pedregal, Parametrized Measures and Variational Principles © Birkhäuser Verlag 1997
162
Chapter 9. Quasiconvexity and Rank-one Convexity
dimension two and show how it dramatically fails. New ideas are needed to clarify the two-dimensional situation. An equivalent way of understanding rank-one convexity that is helpful sometimes is the following. A function W defined on matrices is rank-one convex if and only if the functions of one real variable g(t) = w(Y + tF) are convex for all matrices Y and F with rank (F) = 1. If w is smooth, this is the case if 2
-d2 w(Y
dt
+ tF) I
t=o
2:: 0
for all such Y and F. If this second derivative is written in terms of the derivatives of W we obtain the Legendre-Hadamard condition for rank-one convexity
for all matrices Y E M mxN and vectors a E Rm and n ERN. We have taken here F = a QSl n.
2. Laminates Let us briefly recall (in a slightly different form) how the rank-one convexity condition was introduced in Chapter 1. Let Yi E MmxN, i == 1,2, a E R m and a unit vector n E RN be given in such a way that
(9-1) If Xt is the characteristic function of the interval (0, t) in (0,1) extended by periodicity, the parametrized measure associated to the sequence of gradients
is
(9-2) where n is any bounded domain in RN. Therefore the probability measure 1/ in (9-2) is a gradient parametrized measure (by this we actually mean W1,oo_ parametrized measure) for any t E [0,1] provided the compatibility condition (9-1) holds. Furthermore, by Lemma 8.3, we can assume without loss of generality that Uj - Uy E W~,OO(n), Y = tY1 + (1 - t)Y2 . In this case VUj takes on the values Y1 and Y2 except in small sets Ej , IEjl ----t O. We would like to go
2. Laminates
163
one step further as described in the proof of Lemma 8.6. Assume, in addition to (9-1), that
Y2 = toY?)
+ (1 -
to E (0,1),
to)Y?) ,
yP) - yP) = b0e,
(9-3)
where bERm and e E RN is another unit vector. Let n{ be the part of n where V'Uj = Yi. For j and i fixed, based on the compatibility condition between yP) and
v{i -
Y?),
we can construct a sequence of gradients
{VV{i},
W5,CXJ(n{), whose values essentially alternate between yY) and yP) with preassigned frequency to E (0,1) and normal e to the layers. Let UY2
E
E{i be the set where VV{i does not take either of the two values yP) or yP). Choose k = k(j, i) in such a way that
as j --)
00
uniformly in i = 1, 2. Define
x E n{, else. This sequence {u(j)} is uniformly bounded in w1,CXJ(n) and satisfies u(j) -Uy E W5,CXJ(n). The parametrized measure associated to {Vu(j)} is
(9-4) homogeneous. The probability measure in (9-4) is a gradient parametrized measure provided we have the compatibility conditions (9-1) and (9-3). It is not difficult to generalize this construction when a finite number of matrices is involved if we have the rank-one condition in a recursive way. This basic construction has been referred to as "layers within layers" in the literature and reflects accurately the situation. It motivates the following definition.
Definition 9.1 A set of pairs {(ti' Yi)}l 0, Li ti = 1, Yi E MmxN is said to satisfy the (Hd conditio; if: i) for 1 = 2, rank(Y1 - Y 2 ) :::; 1; ii) for 1 > 2 and possibly after a permutation of indices, rank (Y1 - Y 2 ) :::; 1 and if we set 81
=h
8i=ti+1,
+t2,
Zl
t1
t2
81
81
= -Y1 + -Y2,
Zi=Yi+l,
2:::;i:::;l-1,
the set of pairs {(8i' Zi)}l::;i::;l-l satisfies the (H1-d condition.
164
Chapter 9. Quasiconvexity and Rank-one Convexity
An immediate consequence of our previous discussion is Proposition 9.2
measure v =
If {(ti' Yi)h O. This implies, due to the convexity of U, that the sum
does not converge to O. Since the denominators 0, we conclude
t7
are uniformly distant from
does not converge to 0, either. This is a contradiction with the support of v contained in U. If ti = 0 discard the pair (ti' Y,). In any case Y = t I YI + t2 Y2. We now proceed recursively. By the definition of the (H k ) condition the set of pairs {( Zj) }7E T t verifies some (Hd condition, and therefore we can
sJ,
apply the above procedure to this set of pairs, finding a further subsequence for the indices k and pairs (tli' Yld, i = 1,2, such that rank (Yll - Y12 ) S 1, YI = tll Yll + t12 Yl2 and Y1'i E co(K) provided tli > O. Keep performing these decompositions for Til and subsequent subsets. We clearly see that for
170
Chapter 9. Quasiconvexity and Rank-one Convexity
an appropriate diagonal subsequence and renaming the coefficients ti and the matrices Yi as and y;l, we have
ti
ti1/J(Y;I) - "'" sJ1jJ(Zl) lim lim "'" ~ ~
1--+00 k--+oo
i
= 0,
j
{(tL Y;l) L:::;i:::;l satisfies (HI),
Y;l
E
co(K),
for any continuous function. This implies that the weak limit for
L tioY;l and L sJoz: .
j
•
is the same. This is the end of the proof.
The counterexample is constructed by means of a periodic deformation. Let X = (2Xo - 1) where XO is the characteristic function of (0,1/2) in (0,1) extended by periodicity. We define a deformation U : f! = (0,1)2 C R2 ----; R3 by putting
U1(X)
=
U2(X) = U3(X)
=
1 1
r+
Jo
X1
x1
X(s) ds,
x2
X(s) ds,
X2
X(s
1
+ 4) ds.
The gradient '\lu is the matrix
'\lu(x)
=
(X(~d
t)
X(~2))
t)
X(X1 +X2 + X(X1 +X2 + Notice that the gradient always lies in the three-dimensional subspace, L, of matrices of the form
on~
(9-9)
(x,y,z)
If we consider the sequence of f!-periodic functions Uj(x) = (l/j)u(jx), by the Riemann-Lebesgue lemma (Lemma 8.2) the gradient parametrized measure corresponding to the sequence of gradients {'\lUj} = {'\lu(jx)} is homogeneous and, after the identification suggested in (9-9), given by 1 /J = 16 [0(1,1,1) + 0(1,-1,-1) + o( -1,1,-1) + o( -1,-1,1)]
3
+ 16
[0(1,1,-1)
+ 0(1,-1,1) + 0(-1,1,1) + 0(-1,-1,-1)]
.
The determination of the different weights for /J is a matter of careful counting. Let V denote the set of vertices of the cube C = [-1, 1p, so that supp (/J) = V and the first moment is the origin. We claim that /J is not a laminate. Indeed we have:
171
4. A microstructure that is not a laminate
Proposition 9.10 If f.L is a laminate with supp (11,) = V and first moment 0, 1 /1
=8 [b(U,I) + b(I,-I,-I) + 6(-1,1,-1) + 6(-1,-1,1)] 1
+ 8"
[6(1,1,-1)
+ b(1,-11) + 5(-1,1,1) + 5(-1,-1,-1)]'
The proof reduces to the observation that by Lemma 9.9, in trying to find laminates supported on the set of vertices V and having first moment 0, we can restrict ourselves to the cube C. Since the only rank-one directions in C are the ones given by the axes, the only possibility is the one claimed by the proposition. The situation when the dimension for the target space rn = 2 is drastically distinct. A natural extension of a periodic deformation that might aid in finding a counterexample would be 11, : n = (0,1)2 ----t R2 defined by 111
j
(x) =
.XI
.0
X(s) ds +
lXI +X2 0
X(s
1
+ -) ds, 4
("'2 X(s)ds+ (""+X2 X(s+ ~)d8.
1L2(X) =
Jo
Jo
4
The gradient of 11 lies now in the three-dimensional subspace of 2 x 2 symmetric matrices. The underlying gradient parametrized measure generated from 11, by homogenization can be represented through the identification (
X
+z
z z) ( ) y+z
----t
x,y,z,
and again by I}
1 [b(l,I,I) = 16
+ 1~)
. . + b(-I,l,-l) + 5(-1,-1,1) 1
-t b(l,-I,-I)
[6(1.1,1)
+ 6(1.-1,1) + 6(-11,1) + 6(-1,,1,-1)]
.
What changes now is the set of rank-one directions contained in the corresponding three-dimensional subspace L. Indeed vectors (x, y, z) E L yield rank-one directions if and only if det
(:r: + z z
z
y+z
) = .Ty
+ X z + Yz =
0.
We claim that v is now a laminate. Consider the points in the cube C [-1,1]3 c T, whose coordinates are given below 1 PII = (0,0,0). P1 = (-'2,1,1),
P2 =
1
1
1
(10'-5'-5)'
115 P j = (-Ti' -Ti' -Ti)'
'5
P3 = (1'-7,1),
Pc, = (1,1, -1),
P6 = (-1, - 1, 0) .
172
Chapter 9. Quasiconvexity and Rank-one Convexity
It is easy to check that
are all rank-one (we mean that the coordinates of those differences satisfy xy + yz + xz = 0). We can build a laminate using these directions, supported on the set of vertices of the cube in the following way. Write first
where P2 - PI is a rank-one direction and A E (0,1) is chosen appropriately. Likewise
P2 = A2P3 + (1 - A2)P4 , PI = A3(1, 1, 1) + (1- A3)( -1,1,1), where once again P4 - P3 and (1,1,1) - (-1,1,1) are rank-one directions and A2, A3 E (0,1). Hence
In the same way
P3 = A4(1, 1, 1) + (1 - A4)(I, -1, 1), P4 = A5P5 + (1 - A5)P6 , 1
1
P6 = '2(-1, -1, 1) + '2(-1, -1, -1), where (1,1,1) - (1,-1,1), P5 - P6 , (-1,-1,1) - (-1,-1,-1) are rank-one directions. Use these decompositions to find a laminate supported in V. The laminate, VI, that comes out of this construction is
3
VI
= 16
+
2 16 1
+ 16
(8(-1,-1,-1) (8(1,-1,1) (8(1,1,1)
+ 8(-1,-1,1))
+ 8(-1,1,1))
+ 58(1,1,_1)),
173
5. Rank-one convexity does not imply quasiconvexity By symmetry we can also construct very easily two more laminates, with the origin as first moment and with volume fractions given by
3
1/2
= 16 (8(-1.-1.-1) 2
+ 16 1
+ 16 I/;~
=
3 16
2
+ 16
(8(-1.1.1) (8(1,1,1)
1/2
and
1/3,
+ 8(-1.1.-1))
+ 8(1,1,-1))
+ 58(1.-1.1)) ,
(8(-1,-1,-1) (6(1,l.-1)
+ 8(1,-1,-1))
+ 6(1,-1,1))
+~ (6(1' 11) 16 , + 56(-11 . ,I)) . It is a matter of careful arithmetic to find
Because of the convexity of the set of laminates with the same first moment, is a laminate.
1/
5. Rank-one convexity does not imply quasiconvexity Even though the arguments in the preceding section are convincing they may be confusing to readers with not much experience with laminates. The notation might also be somewhat confusing. We intend in this section to give an explicit counterexample of a rank-one convex function that is not quasieonvex. The reader will immediately see the connection with the example in the last section. Indeed, we are going to produce a rank-one convex function W such that
w(O) > (w, 1/) where
1
1/
= 16
[6(l.1,1)
+ 136
+ 6(1,-1,-1) + 6(-1,1,-1) + 6(-1,-1,1)]
[6(1,1,-1)
+ 8(1,-1.1) + 6(-1,1,1) + 6(-1,-1,-1)]
,
in the context of the previous section for Tn = 3. Therefore W cannot be quasiconvex and 1/ cannot be a laminate. Rather that dealing with 1/ directly we will use the periodic deformation, u: rl = (0,1)2 C R2 ----t R:3 that generates it
u(x) =
(rl X(s)ds, r .fa .fo
2
X(s)ds,
r
.fa
1
+
X2
X(s+
~)dS), 4
174
Chapter 9. Quasiconvexity and Rank-one Convexity
where X = (2Xo - 1) and XO is the characteristic function of (0,1/2) in (0,1) extended by periodicity. Recall that the gradient of u is given by
V'u(x)
=
and its image lies in the three-dimensional subspace of 3 x 2 matrices in (9-9). The first lemma gives a reformulation of quasiconvexity in terms of periodic deformations. It is very convenient in this form. Lemma 9.11 only if
A continuous function rp : M mxN
r
i(O,l)N
rp(Y
+ V'u(x)) dx 2
--+
R is quasiconvex if and
rp(Y),
for every matrix Y and every smooth, periodic deformation
(9-10) U :
RN
--+
Rm.
Proof. If (9-10) holds for every periodic deformation u then rp is trivially quasiconvex since any U E W~·CXJ(o) for 0 = (0, l)N can be extended by periodicity. Assume now that rp is quasiconvex and let u be any periodic deformation. Consider the sequence Uj(x) = Yx + (l/j)u(jx), x E O. V'Uj = Y + V'u(jx) and by the Riemann-Lebesgue lemma, Lemma 8.2, the gradient parametrized measure associated to {V'Uj} is homogeneous and given by (rp, VI = (Notice that
Inl = 1.)
L
rp(Y
1
rp(Y
+ V'U(x)) dx,
Hence, if rp is quasiconvex, Jensen's inequality
+ V'u(x)) dx 2
rp
(Y + LV'u(x) dX) = rp(Y)
should hold. We have used the fact that if u is periodic then the integral of its gradient over one period cell vanishes. • The basic ingredient for our counterexample is the cuhic polynomial '1/) defined on the subspace L, through the identification in (9-9), by
1j;(x, y, z) = xyz. Because the only rank-one directions contained in L are the directions given by the axes, 1j; is certainly rank-one convex in this subspace (indeed rank-one affine). Moreover, it is easy to check
inr 1j;(V'u(x))dx = -~8 < 0 = 1j;(O,O,O).
(9-11)
Our task is to extend 1j; to all of the space of 3 x 2 matrices so that it remains rank-one convex and the inequality (9-11) is preserved. Let M stand for the space of 3 x 2 matrices and 7f for the orthogonal projection onto L. For a matrix X E M, IXI is the usual euclidean norm.
175
5. Rank-one cOIlYexit.y does not. imply quasiconvexity
Lemma 9.12
For ( > 0 fixed, there exists k = k( E) > 0 such that the function
W(X) =I/J (7fXr) +E IXI 2 +E IXI ,1 +klX -7fXI 2
(9-12)
is rank-one convex on M.
Proof. Since W is smooth, we have t.o check that 2
~W(Y d
dt
+ tF) I
::;> 0
t=o
holds for every mat.rix Y and every rank-one convex matrix F. We first compute 2
-d2 W(Y
dt
+ tF) I
t=o
(P = -2~J(7fY + t7fF) I
t=u
dt
(9-13)
+ 2E 1F12 + 4E 1Y121F12 + SE(Y . F)2 +2klF-7fFI 2 ,
where F . Y is the dot prod uet for matrices. The function J/l (7f X) is a third degree, homogeneous polynomial so that we can find c > 0 with
for all matrices Y, F. From (9-13) we have
for all matrices Y, F. In particular if
IYI
::;> c:/(4E),
(9-14) for all F. On the other hand we also see from (9-13) that
2 d 2 W(Y + tF) I
dt
f.~()
::;> (P2 VJ( 7fY + t7f F) I
dt
+ 2f 1F12 + 2k IF - 7f F12, (9-15)
f=()
for all Y. F. Let h(Y, F. k) denote the right-hand side in (9-15) for h is a continuolls [unction of all it:,) arguments. If we let
K =
{(y. F) EM x M : IYI : :.;
;~,
rank (F)
=
1,
E
IFI = I} ,
> 0 fixed.
176
Chapter 9. Quasiconvexity and Rank-one Convexity
a compact set of pairs of matrices, we claim that ko can be found so that h(Y, F, ko) > E for all pairs (Y, F) E !C. Otherwise we could find a sequence (Yk, F k ) E !C such that h(Yk , Fk, k) =::: E. If, by compactnes8 of !C, (Yk, Fk) ~ (Y, F) E K as k ~ 00, it follows by (9-15), that F = 7rF ELand
This contradicts the fact that 'ljJ is rank-one convex in L. Hence there exi8ts ko such that
-2W(Y + tF) I > E d dt t=O for each pair (Y, F) E K. This, together with (9-14), proves the lemma. 2
•
The following theorem is a direct consequence of the previous results. Theorem 9.13 There exist E > 0 and k > 0 such that W given in (9-12) is rank-one convex but not quasiconvex. Proof. For the periodic deformation u considered above, we can choose E > 0 sufficiently small so that by (9-11) and because \1u is uniformly bounded,
r
leo,1)2
('ljJ(\1u)
+ EJ\1uJ 2 + EJ\1uJ 4)
dx
0, choose 8 > 0, so that
ii) Once 8 > 0 is fixed, since u k \lT/8 ~ 0 in L2(0), the linear functionals
183
2. Technical preliminaries
converge strongly to 0 in H- 1 (0,); therefore, there is a ko, such that if k ~ ko
I(Tk , rp)1 :s;
unif. in rp.
f,
iii) There is a k1' such that if k ~ kl
Hence if k ~ max(ko, k 1 ),
• Another elementary device we need in this context is the average formula which we now establish as a lemma. It yields the homogenized version of any parametrized measure with the additional property on the divergences.
Assume that uk --"'. ,. u in Loo(O,), where u E RN is constant, and divu k - t 0 in H- 1(0,). Let v = {VX LE!1 be the parametrized measure associated to {Uk} and define
Lemma 10.5
There exists {ud such that Uk --"'. ,. u in Loo(O,), divUk associated parametrized measure is v.
-t
0 in H-1(0,), and the
Proof Given j, let 7]j be a cut-off function for 0,:
o :s; 7]j :s; 7]1
1,
= 1, if dist(x, 00,) 7]j
= 0 on 00"
1V'7]jl
~
1
-:, J
:s; Cj.
For kEN, {a + En: a E 0" E < k- 1 } is a covering of 0,. By the Vitali's covering lemma there exists a countable family {ai, Ed and a null-measure set N, such that
Define
uk,j (x) = { u + (uk ( X~ ai ) - u) 7]j (x ~i ai ), x E ai + fin u,
otherwise.
Chapter 10. Analysis of Divergence-Free Parametrized Measures
184
For
(y)
= L EfV'dy.
Similar to the proof of the previous lemma, given j, we can choose kj, so as to make the two terms on the right-hand side of (10-2) arbitrarily small. If
u j = ukj,j,
divuj
-->
0 in H- 1 (0).
Now for ~ E C(O) and cp E C(RN),
1 o
cp (u j (x))
~ (x)
dx = L
i
Ef
1 0
cp ((u kJ (y) - u) T/j (y)
=LEf~(ai) L 0,
4. Characterization of divergence-free parametrized measures Lemma 10.8
187
For u E co(K), the set of probability measures
Au = {v E C(B)': suppv C K, uk: 0
----+
i
= u,
>..dv
K, divu k
----+
v is associated to {uk},
0 in H-1(0)}
is convex and weak-* closed.
Proof The weak * closedness part is easy using appropiate subsequences since C(B) and Ll(O) are separable. Let Vi E Au, >.. E (0,1) and uf ~ u in LOO(O), divuf ----+ 0 in H-l(O),i = 1,2. Take
D C 0, smooth with IDI = >..101.
Apply Lemma 10.4, and have
For uk = u~ + XD (u~ - m~), apply Lemma 10.5 and let the corresponding • parametrized measure for {uk} be >"Vl + (1 - >")V2. Theorem 10.9
Au = {v
E
C(B)' : v is a probability measure, suppv
L
>..dv(>..) = u} .
Proof Assume that rp
E
C(B) is such that (rp, v) 2:> 0,
'v'v E Au.
Then
whenever u j ~ u in LOO(O),
divu j so that
----+
0 in H-1(0),
(rp)diV (u) 2:>
o.
Thus, by the final remark in the previous section,
(rp)** (u) = (rp)div (u) 2:> 0,
C
K,
188
Chapter 10. Analysis of Divergence-Free Parametrized Measures
and for any /-l, probability measure with first moment u and support in K
By Hahn-Banach,
• We now prove a nonhomogeneous version of Theorem 10.9 which consists in "patching" sequences appropriately. This is the natural argument when going from the homogeneous version of some fact to the non-homogeneous as we have already done before.
Theorem 10.10 Let v = {v x } xEO be a family of probability measures with supp(vx ) C K and let
u(x) = There exists a sequence {u j such that
},
L
Advx (>\) E LOO(D).
uj
Inrcp(x, uj(x)) dx
:
-+
D
-+
K, divu j
-+
divu in H- 1 (D) and
r r cp(x,A)dvx(A)dx,
JO~N
for every Caratheodory function cp. Proof. Let N be the complement of the set of the points a E D where
(1
lu(x) - u(a)12 dX)1/2
-+
0,
a+En
Let rda)
E
-+
O.
> 0, such that if E < rk(a) then
1
21 lu(x) - u(a)1 dx =:: 2 ' a+En k
and consider a countable set of products ((X)1jJ(A) whose linear combination are dense in Ll(D;C(EM)) where as before EM = {IAI =:: M} eRN and K C EM. Apply Lemma 7.9 and find {akJ, {EkJ with the appropriate properties. Choose a sequence r]k of smooth cut-off functions such that r]k
=
1 in Dk
= { xED:
dist(x, aD)
r]k = 0 on aD,
1\7r]kl =:: 2k.
~ ~}
,
4. Characterization of divergence-free parametrized measures
Let {u~} generate vx , with u~ : n --+ K, div u~ LOO(n). Define the functions uk,j by
~) U~ki . (x-aki) ~ { T/k ( Eki u(x), For rp E HJ(n),
IIV'rpll£2(O)
+ (1 -
--+
189
0 in H- 1 (n) and u~ ...."'.. 0 in
ki )) u (aki, ) T/k (x-a ~
x E aki + Eki n otherwise.
:S. 1,
10 (u(x) - uk,j(x)) V'rp(x) dx = ~ lki+ Ekif! (u(x) - u(aki)) V'rp(x) dx + L 10 (u(aki) - ULi (y)) Y' (T/krpki) dy " - L, if!r (u(aki)-uL(Y))Y'T/k rpkidy where
and (Xki are constants to be chosen in a moment. Observe that
and (Xki is chosen so that, by Poincare's inequality, the H1-norm of all such rpki are uniformly bounded in Hl(n). We now proceed to estimate the three terms h, Ih and IIIk.
Ihl
:S.
L
:S.
kL
IIV'rpll£2(aki+Ekif!)
1
Ilu -
u(aki)II£2(aki+ E kif!)
IIV'rpll£2(aki+ Eki O )
i
1
