URL:http://www.emath.fr/cocv/

**LINKS BETWEEN YOUNG MEASURES ASSOCIATED**
**TO CONSTRAINED SEQUENCES**

## Anca-Maria Toader

^{1}

**Abstract**. We give necessary and sufficient conditions which characterize the Young measures
associated to two oscillating sequences of functions,*u**n*on*ω*1*×**ω*2and*v**n*on*ω*2satisfying the constraint
*v**n*(y) = _{|}_{ω}^{1}

1*|*

R

*ω*_{1}*u**n*(x, y)dx. Our study is motivated by nonlinear effects induced by homogenization.

Techniques based on equimeasurability and rearrangements are employed.

**AMS Subject Classification.** 28, 49.

Received February 8, 2000. Revised July, 2000.

## 1. Introduction

Given two sequences of functions,*u**n*on*ω*1*×ω*2and*v**n*on*ω*2such that the constraint*v**n*(y) = 1

*|ω*1*|*
Z

*ω*1

*u**n*(x, y)dx
holds, one wants to characterize the relationship between their associated Young measures.

Our main result (Th. 3.1) gives necessary and sufficient conditions for the above problem in terms of
distribution measures and decreasing rearrangements. It can be interpreted as follows: Young measures cap-
ture something of the oscillating behaviour of the sequence (u*n*) and the integration

Z

*ω*_{1}

*u**n*(x, y)dx destroyes
part of the oscillations of*u**n*. We use Young measure techniques introduced by Ball, Tartar, Balder, Valadier.

Notions and results from Probability Theory are employed; particularly, the order relation *≺* on the set of
positive measures due to Choquet and Loomis, and related results (see Cartier *et al.*[4] and Meyer [6]). The
corresponding preorder relation*≺*on the set of nonnegative *L*^{1} functions introduced by Hardy *et al.* is used,
as well as properties of doubly stochastic operators proved by Ryff.

Our study was motivated by more complex questions in relation with nonlocal effects induced by homogenization.

Consider the degenerate elliptic equation studied by Amirat*et al.* in [1]:

*∂*

*∂x*

*a**n*(x, y)*∂u**n*

*∂x* (x, y)

=*f*(x, y) in ]0,1[*×*]0,1[,
*u**n*(0, y)=*u**n*(1, y) = 0 on ]0,1[.

(1.1)

By homogenization a nonlocal effect appears expressed in terms of a kernel. We introduce a parameter *γ*
(following an idea of Tartar [10]) by setting*a**n*(x, y) := *a** _{−}*(x, y)/(1 +

*γb*

*n*(x, y)) where 1/a

*n*

***1/a

*for the*

_{−}*Keywords and phrases:*Young measures, rearrangements, doubly stochastic operators.

1Faculdade de Ciˆencias, CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal;

e-mail:amtan@lmc.fc.ul.pt

c EDP Sciences, SMAI 2000

weak*∗*topology and *b**n* :=*a*_{−}*/a**n**−*1. Then the above problem corresponds to the case*γ*= 1. The solution
*u**n* writes in series of powers of*γ*like*u**n,γ*(x, y) =*v*_{n}^{0}(x, y) +*v*_{n}^{1}(x, y)γ+*· · ·*+*v*_{n}* ^{k}*(x, y)γ

*+*

^{k}*. . .*and the analysis would be based on analyticity properties in the parameter

*γ. The kernel describing the nonlocal effect is the*weak limit of

*K**n,γ*(x, ξ, y) = *b**n*(ξ, y)
1 +*γ*R1

0 *b**n*(x, y)dx
Z 1

0

*b**n*(x, y)dx*−b**n*(x, y)

*.* (1.2)

One would like to characterize the nature of kernels that may appear as weak limits of*K**n,γ* in (1.2), in order
to understand the constitutive laws of the homogenized materials. This is still an open problem.

We inscribe our contribution in the effort to characterize such kernels by solving the symplified problem presented in the beginning.

In Section 2 we make a brief recall on notions to be employed: distribution measures, equimeasurability,
rearrangements, the relation *≺*, Young measures.

Section 3 is dedicated to our main result: we state it and give its proof by making use of auxiliary results contained in Section 4.

## 2. Preliminary notions

To describe precisely the type of questions we are considering here, we begin by recalling a few facts about
distribution measures and equimeasurable functions. Let Ω be a bounded domain in R* ^{N}*, denote by

*L*

*the Lebesgue measure onR*

^{N}*and by*

^{N}*B*(Ω) the Borel

*σ-field of Ω. Let*

*f*: Ω

*→*R

^{+}be a nonnegative, measurable function; we represent by

*µ*

*f*its

*distribution measure*defined by

*µ**f*(B) :=*L**N*(f^{−}^{1}(B)), *∀B∈ B*(R+).

We denote by*f** ^{∗}* the

*decreasing rearrangement*of

*f*given by:

*f** ^{∗}*(s) := sup

*{t >*0 :

*µ*

*f*(]t,

*∞*[)

*> s}·*

It is easily checked that*f** ^{∗}* is the unique (up to modifications on Lebesgue zero-measure sets) nonincreasing
function on [0,

*L*

*(Ω)[ such that*

^{N}*f*

*and*

^{∗}*f*have the same distribution measure

*µ*

*f*. Finally, we will say that

*f, g≥*0 are equimeasurable or, equivalently, that

*f*is a rearrangement of

*g*if they have the same distribution measure. We will denote this equivalence relation by

*f*

*∼g.*

Hardy*et al.* introduced (see [5]) the following preorder relation on the set of nonnegative functions in*L*^{1}(Ω):

for*f* and*g* in*L*^{1}(Ω)

*g≺f* iff
Z

Ω

*F*(g(t))dt*≤*Z

Ω

*F*(f(t))dt ,

for all convex, continuous functions*F* :R^{+} *→*R. Observe that *f* *≺g* and *g* *≺f* is equivalent to*f* *∼g. We*
recall from [5] the following property of the preorder relation between functions in*L** ^{∞}*(Ω), in terms of decreasing
rearrangements

*g≺f* iff

Z *t*
0

*g** ^{∗}*(s)ds

*≤*Z

*t*0

*f** ^{∗}*(s)ds

*∀t∈*[0,

*L*

*(Ω)[, Z*

^{N}

_{L}*N*(Ω)

0

*g** ^{∗}*(s)ds=

Z _{L}*N*(Ω)
0

*f** ^{∗}*(s)ds.

(2.1)

Actually we shall only use functions which take values in a compact interval.

On the set of positive finite measures, the corresponding order relation was introduced by Choquet and, in a different framework, by Loomis (see [4] and [6]).

**Definition 2.1.** *Two positive finite measures on* ([α, β],*B*([α, β])), *ν* *and* *µ, having the same total mass*
*(µ([α, β]) =ν([α, β])) satisfy* *µ≺ν* *if and only if*

Z *β*
*α*

*φ(z)dµ(z)≤*Z *β*
*α*

*φ(z)dν(z),*
*for all continuous and convex functionsφ*: [α, β]*→*R*.*

**Remark 2.2.** Given two positive functions such that *g* *≺* *f*, their distribution measures *µ**g* and *µ**f* satisfy
*µ**g**≺µ**f*. Conversely, given two measures that satisfy*µ≺ν, ifg*and*f* have the distribution measures*µ*and*ν*
respectively, then*g≺f*.

The “old” tool of Young measures has proven crucial for applications in asymptotic analysis. Young measures techniques were developed by Tartar [9], Ball [3], Balder [2], Valadier [11], etc.

In the sequel we make a brief recall on Young measures following the notations and framework from [11].

We call*Young measure*any positive measure*µ*on Ω*×S* (S is a metrizable space) whose projection on Ω is
*L** ^{N}*. Let

*Y*(Ω

*×S) be the set of all Young measures on Ω×S.*

We will not distinguish*µ*from its disintegration (µ*x*)*x**∈*Ωwhich is a measurable family of probabilities on*S*
such that for any*ψ* : Ω*×S→*R,*µ-integrable,*

Z

Ω*×**S*

*ψdµ*=
Z

Ω

Z

*S*

*ψ*(x, ξ)*dµ**x*(ξ)d*L** ^{N}*(x).

For each measurable function *a* : Ω*→* *S* we associate a Young measure *µ**a*, with support in the graph of*a,*
defined by

*hµ**a**, φi*=
Z

Ω*×**S*

*φ(x, λ)dµ**a*(x, λ) =
Z

Ω

*φ(x, a(x))dx,*
for all positive Carath´eodory integrands*φ*: Ω*×S* *→*R.

On*Y*(Ω*×S) we consider thenarrow* topology,*i.e., the weakest topology that makes continuous the maps*
*µ7→*Z

Ω*×**S*

*φ(x, λ)dµ(x, λ),*
for all bounded Carath´eodory integrands*φ.*

**Remark 2.3.** 1) The set of Young measures associated to a sequence of functions uniformly bounded
in*L*^{1}(Ω;R* ^{d}*) is relatively compact in

*Y*(Ω

*×*R

*).*

^{d}2) If*S*is a metrizable compact space, then every set of Young measures associated to a sequence of measurable
functions *a**n*: Ω*→S* is relatively compact in*Y*(Ω*×S).*

**Remark 2.4.** 1) Given a uniformly bounded sequence (a*n*) in*L*^{1}(Ω;R* ^{d}*), using Remark 2.3 1), we may assume
that, up to a subsequence of (a

*n*), the sequence of their associated Young measures, narrow converges to some

*µ*= (µ

*x*)

*x*

*∈*Ω

*∈ Y*(Ω

*×*R

*).*

^{d}2) For a sequence of functions*a**n* : Ω*→S* we say that *µ* is the Young measure associated to the sequence
(a*n*), or, that (a*n*) gives rise to the Young measure*µ, if the Young measures associated to (a**n*) narrow converge
to*µ,i.e., for all bounded Carath´*eodory integrand*φ,*

Z

Ω

*φ(x, a**n*(x))*dx→*
Z

Ω*×**S*

*φ(x, λ)dµ(x, λ).*

## 3. Main result

Theorem 3.1 below was conjectured by Tartar in some discussions we had together, during his visits to Lisbon, in December 1998.

Consider two domains*ω*1*⊂*R* ^{N}* and

*ω*2

*⊂*R

*and a compact interval [α, β] inR. We may assume without losing generality that*

^{M}*|ω*1

*|*=

*|ω*2

*|*= 1.

**Theorem 3.1.** *Given are the sequences* *u**n* : *ω*1 *×ω*2 *→* [α, β] *and* *v**n* : *ω*2 *→* [α, β] *such that* *v**n*(y) =
R

*ω*_{1}*u**n*(x, y)dx. Assume that the sequence(u*n*)*gives rise to the Young measureν* = (ν*x,y*)(x,y)*∈**ω*_{1}*×**ω*_{2} *and that the*
*sequence*(v*n*)*gives rise to the Young measureµ*= (µ*y*)*y**∈**ω*_{2}*. One defines a functionf* :*ω*1*×ω*2*×*(0,1)*→*[α, β]

*such thatf*(x, y,*·*)*is nonincreasing and its distribution measure isν**x,y**, and similarly, a functiong*:*ω*2*×*(0,1)*→*
[α, β] *such thatg(y,·*)*is nonincreasing and its distribution measure isµ**y**. Then one has*

Z *t*
0

*g(y, s)ds≤*
Z

*ω*_{1}

Z *t*
0

*f*(x, y, s)dsdx, (3.1)

*with equality fort*= 1.

*Conversely, iff* *andg* *satisfy the above inequality for all* *t∈*[0,1) *and the corresponding equality fort*= 1,
*then, denoting by* *ν**x,y* *and* *µ**y* *the distribution measures of* *f*(x, y,*·*) *and* *g(y,·*), respectively, there exists a
*sequence* *u**n* : *ω*1*×ω*2 *→* [α, β] *that gives rise to the Young measure* *ν* *and the sequence defined by* *v**n* :=

R

*ω*1*u**n*(x, y)dx*gives rise to the Young measure* *µ.*

The author presented a particular case of the above result (the Young measure*ν* = (ν*x,y*)(x,y)*∈**ω*1*×**ω*2 did not
depend on*x:* *ν**x,y*=*ν**y*) at the Equadiff99 conference held in Berlin.

*Proof.* For the direct implication let*t∈*[0,1] arbitrarily fixed. Then by Lemma 4.5 there exists a subsequence
*v**n**k* of*v**n* and a sequence of characteristic functions*χ*^{v}* _{k}*:

*ω*2

*→ {*0,1

*}*,

*χ*

^{v}

_{k}** t*such that

*χ*^{v}_{k}*v**n*_{k}***
Z *t*

0

*g(y, s)ds.*

On the other hand, applying Lemma 4.5 with the corresponding subsequence (u*n**k*) and with the same number
*t, we obtain that for any sequence of characteristic functions* *χ**k* :*ω*1*×ω*2*→ {*0,1*}*such that *χ**k* ** t, up to a*
subsequence, we have that

weak lim

*k* *χ**k**u**n*_{k}*≤*Z *t*
0

*f*(x, y, s)ds, (3.2)

and in particular the above inequality holds for*χ*^{v}* _{k}*. Then
Z

*t*

0

*g(y, s)ds*= weak lim

*k* *χ*^{v}_{k}*v**n** _{k}* = weak lim

*k* *χ*^{v}* _{k}*
Z

*ω*_{1}

*u**n** _{k}*(x, y)dx

= weak lim

*k*

Z

*ω*1

*χ*^{v}* _{k}*(y)u

*n*

*(x, y)dx*

_{k}*≤*Z

*ω*1

Z *t*
0

*f*(x, y, s)dsdx,

and the direct implication turns out, for*t* *∈*[0,1[. For*t* = 1,*χ*^{v}_{k}*→* 1 strongly and (3.2) holds with equality
which yields equality in (3.1).

Conversely, suppose that *f* and *g* satisfy (3.1) for *t* *∈*[0,1) and the corresponding equality for *t*= 1. We
construct a sequence*u**n* giving rise to the Young measure *ν* and such that the sequence*v**n* :=R

*ω*1*u**n*(x, y)dx
gives rise to the Young measure *µ*as follows: by property (2.1), (3.1) is equivalent to*g(y,·*)*≺*R

*ω*_{1}*f*(x, y,*·*)dx.

Applying Lemma 4.4 it turns out that there exists a positive measure*θ*on [0,1]*×*[0,1] whose both projections

are the Lebesgue measure and such that
*g(y, t) =*

Z 1 0

Z

*ω*1

*f*(x, y, s)dxdθ*t*(s).

Here*θ**t*is the disintegration of*θ*with respect to the Lebesgue measure. Define*u*:*ω*1*×ω*2*×*[0,1]*×*[0,1]*→*[α, β]

by*u(x, y, s, t) :=f*(x, y,Φ*t*(s)), where Φ*t*: [0,1]*→*[0,1] has the distribution measure*θ**t*. The existence of Φ*t*is
ensured by Remark 4.1. Then*g* writes

*g(y, t) =*
Z 1

0

Z

*ω*1

*f*(x, y,Φ*t*(s))dxds=
Z 1

0

Z

*ω*1

*u(x, y, s, t)dxds.* (3.3)

One takes*u**n*(x, y) :=*u(x, y,*[nx1],[ny1]) (here [z], with*z∈*R, represents the fractional part of*z*and, by*x*1, we
represent the first component of*x). By Riemann-Lebesgue theorem it turns out thatu**n*gives rise to the Young
measure *ν. By a similar argument, the sequence* *w**n* defined by *w**n*(y) :=*g(y,*[ny1]) gives rise to the Young
measure *µ. Having in mind the definition of the sequencev**n*(y) :=R

*ω*1*u**n*(x, y)dx, it turns out that *v**n**−w**n*

converges uniformly to 0, and since*w**n* gives rise to the Young measure *µ, so does the sequencev**n*, and the

proof is complete.

## 4. Auxiliary results

**Remark 4.1.** Given a finite positive measure*µ*on [α, β],*µ*:*B*([α, β])*→*R^{+}, there exists a measurable function
*f* : [0, µ([α, β])]*→*[α, β] that transports the Lebesgue measure on [0, µ([α, β])] into *µ,i.e.* *µ(B) =L*^{1}(f^{−}^{1}(B))
for all*B* in *B*([α, β]). We can take for instance the nonincreasing function that is given by

*f*(x) := sup*{t∈*[α, β] :*µ([t, β])> x}·*

The following two properties are to be used in the proof of Lemma 4.4.

**Property 4.2.** *Consider a finite positive measureθon* Ω*×*Ω^{0}*with proj*_{Ω}*0**θ*=*ν. If the measureν* *is absolutely*
*continuous with respect to a positive measure* *l,* *dν(y) =* *τ(y)dl(y)* *for a function* *τ* *∈* *L*^{1}* _{l}*(Ω

*), then*

^{0}*θ*

*may be*

*disintegrated with respect tol*

*according to the formula*

*dθ(x, y) =θ*^{l}* _{y}*(x)dl(y),

*whereθ*^{l}* _{y}*(x) =

*τ(y)θ*

*y*(x)

*and*(θ

*y*)

*y*

*∈*Ω

*0*

*is the disintegration ofθ*

*with respect toν.*

Let (Ω, m) and (Ω^{0}*, l) be two measure spaces with the same total mass (m(Ω) =l(Ω** ^{0}*)

*<*+

*∞*),

*m*and

*l*being positive measures.

**Property 4.3.** *If* *f* : Ω*→*R*andg* : Ω^{0}*→*R*are integrable functions such thatf*(x)*< g(y)* *for all* *x∈*Ω *and*
*y∈*Ω^{0}*, then*

Z

Ω

*f*(x)dm(x)*<*

Z

Ω^{0}

*g(y)dl(y).*

**Lemma 4.4.** *Given two measurable functions* *u*: Ω *→*[α, β] *and* *v* : Ω^{0}*→*[α, β] *such that for all continuous*
*and convex functions*Φ : [α, β] *→*R*,*

Z

Ω^{0}

Φ(v(y))dl(y)*≤*Z

Ω

Φ(u(x))dm(x) *holds. There exists then a positive*
*measureθ, on*Ω*×*Ω^{0}*, whose projections on*Ω*and*Ω^{0}*aremandl, respectively, and such thatl-almost everywhere*
*in* Ω^{0}

*v(y) =*
Z

Ω

*u(x)dθ**y*(x),
*where*(θ*y*)*y**∈*Ω^{0}*is the disintegration of* *θwith respect tol.*

This result is a consequence of the results obtained by Ryff in [7] and [8] regarding equivalent characterizations,
in terms of doubly stochastic operators, for the preorder relation*≺*between functions. Lemma 4.4 is also very
close to Cartier’s theorems stated in terms of dilatations (see Ths. 1 and 2 in [4] and Ths. 35 and 36 in [6],
p. 288). We shall give a direct (sketched) proof.

The proof is to be done in 3 steps.

**Step 1.**

Denote by Θ the set of positive measures*θ* on Ω*×*Ω* ^{0}* such that the projections of

*θ*on Ω and Ω

*are*

^{0}*m*and

*l,*respectively. Θ is a subset of the linear space of measures on Ω

*×*Ω

*. The set Θ is compact with respect to the weak topology of measures.*

^{0}Let us define for each*θ∈*Θ the function*w(y) =*R

Ω*u(x)dθ**y*(x) where (θ*y*)*y**∈*Ω* ^{0}* is the disintegration of

*θ*with respect to

*l. Note that the dependency ofw*on

*θ*is linear. Let us introduce the functional

*J*: Θ

*→*R

^{+},

*J*(θ) =
Z

Ω*0*(v*−w)*^{2}*dl(y).*

*J* depends on *θ* through *w, and, due to the lower semicontinuity of the norm in* *L*^{2}* _{l}*(Ω

*), it turns out that*

^{0}*J*is lower semicontinuous. Since Θ is compact,

*J*attaines its minimum,

*i.e.*there exists

*θ*

^{0}

*∈*Θ such that

*J*(θ

^{0}) = inf

*θ*

*∈*Θ

*J(θ). Denote by*

*w*0(y) :=R

Ω*u(x)dθ*^{0}*y*(x) the function corresponding to*θ*^{0}. We shall
prove that*w*0=*v.*

**Step 2.**

*θ*^{0} has the following property: if*A*1 and*A*2 are subsets of Ω such that
*u(x*1)*< u(x*2) for all*x*1*∈A*1*, x*2*∈A*2

and if*B*1and*B*2are subsets of Ω* ^{0}* such that

*v(y*1)*−w*0(y1)*< v(y*2)*−w*0(y2) for all*y*1*∈B*1*, y*2*∈B*2*,*

then, either*θ*^{0}(A1*×B*2) = 0 or*θ*^{0}(A2*×B*1) = 0. Otherwise, if*θ*^{0}(A1*×B*2)*>*0 and*θ*^{0}(A2*×B*1)*>*0 one can
increase*θ*^{0}in the sets*A*1*×B*1and*A*2*×B*2and decrease*θ*^{0}in the sets*A*1*×B*2and*A*2*×B*1by maintaining the
same projections *m* and*l* on Ω and Ω* ^{0}*, respectively, and obtain a smaller value for

*J*. Indeed, assuming that

*θ*

^{0}(A1

*×B*2)

*>*0 and

*θ*

^{0}(A2

*×B*1)

*>*0 we can choose two measures on Ω

*×*Ω

*,*

^{0}*θ*

^{12}and

*θ*

^{21}such that suppθ

^{12}

*⊂*

*A*1

*×B*2,

*θ*

^{12}

*≤θ*

^{0}

*|*

*1*

^{A}*×*

*B*

_{2}, suppθ

^{21}

*⊂A*2

*×B*1,

*θ*

^{21}

*≤θ*

^{0}

*|*

*2*

^{A}*×*

*B*

_{1}and

*θ*

^{12}(A1

*×B*2) =

*θ*

^{21}(A2

*×B*1) =

*γ >*0.

Denote *µ*1 := proj_{Ω}*θ*^{12} (suppµ1 *⊂* *A*1), *ν*2 := proj_{Ω}*0**θ*^{12} (suppν2 *⊂* *B*2), *µ*2 := proj_{Ω}*θ*^{21} (suppµ2 *⊂* *A*2),
*ν*1:= proj_{Ω}*0**θ*^{21}(suppν1*⊂B*1). Note that*µ*1(A1) =*ν*2(B2) =*µ*2(A2) =*ν*1(B1) =*γ.*

Let us define*θ*^{11}:= ^{1}_{γ}*µ*1*⊗ν*1, *θ*^{22}:=_{γ}^{1}*µ*2*⊗ν*2and note that suppθ^{11}*⊂A*1*×B*1 and suppθ^{22}*⊂A*2*×B*2.
Consider*θ** ^{t}*:=

*θ*

^{0}+t θ

^{11}

*−θ*

^{12}

*−θ*

^{21}+

*θ*

^{22}

. For*t∈*[0,1] the measure*θ** ^{t}*is positive since

*θ*

^{12}

*≤θ*

^{0}and

*θ*

^{21}

*≤*

*θ*

^{0}on the disjoint sets

*A*1

*×B*2and, respectively,

*A*2

*×B*1. Calculating the projection of

*θ*

*on Ω we have for every*

^{t}*A∈ B*(Ω) that

*θ*

*(A*

^{t}*×*Ω

*) =*

^{0}*θ*

^{0}(A

*×*Ω

*) +*

^{0}*t θ*

^{11}(A

*×*Ω

*)*

^{0}*−θ*

^{12}(A

*×*Ω

*)*

^{0}*−θ*

^{21}(A

*×*Ω

*) +*

^{0}*θ*

^{22}(A

*×*Ω

*)*

^{0}=*m(A) +*
*t*

1

*γ**µ*1(A)ν1(Ω* ^{0}*)

*−θ*

^{12}(A

*×B*2)

*−θ*

^{21}(A

*×B*1) +

_{γ}^{1}

*µ*2(A)ν2(Ω

*)*

^{0}=*m(A)+t*
1

*γ**µ*1(A)ν1(B1)*−µ*1(A)*−µ*2(A)+

1

*γ**µ*2(A)ν2(B2)

= *m(A), that is proj*_{Ω}*θ** ^{t}* =

*m*and by similar arguments we obtain also that proj

_{Ω}

*0*

*θ*

*=*

^{t}*l.*

Therefore*θ*^{t}*∈*Θ for *t∈*[0,1].

Let us calculate*J(θ** ^{t}*) in order to evaluate the derivative

_{dt}

^{d}*J*(θ

*)*

^{t}*|*

*. Let us make first some remarks useful to the computations.*

^{t=0}*θ*

^{21}

*≤*

*θ*

^{0}on

*A*2

*×B*1, consequently, their projections satisfy

*ν*1

*≤*

*l*and therefore

*ν*1

is absolutely continuous with respect to *l. According to Property 4.2 we can disintegrate* *θ*^{11} and *θ*^{21} with
respect to*l. Similar arguments employed with the measureν*2 permit us to conclude that*θ*^{12} and*θ*^{22} may be
disintegrated with respect to*l. In the following, by (¯θ*^{11}* _{y}* ), (¯

*θ*

^{12}

*), (¯*

_{y}*θ*

^{21}

*) and (¯*

_{y}*θ*

^{22}

*), we denote the disintegrations*

_{y}with respect to*l* of*θ*^{11},*θ*^{12},*θ*^{21}and*θ*^{22}, respectively.

*J*(θ* ^{t}*) =
Z

Ω^{0}

(v(y)*−*
Z

Ω

*u(x)dθ*_{y}* ^{t}*(x))

^{2}

*dl(y)*

= Z

Ω*0*

*v(y)−w*0(y)*−t*
Z

Ω

*u(x)dθ*¯^{11}*y* (x)*−*
Z

Ω

*u(x)dθ*¯^{12}*y* (x)

*−*
Z

Ω

*u(x)dθ*¯_{y}^{21}(x) +
Z

Ω

*u(x)dθ*¯_{y}^{22}(x)
2

*dl(y).*

Since in the above expression the integrand is a polynomial of degree 2 in*t, we obtain that*
*d*

*dtJ*(θ* ^{t}*)

*|*

*= 2 Z*

^{t=0}Ω^{0}

(v(y)*−w*0(y))
Z

Ω

*u(x)dθ*¯_{y}^{11}(x)*−*Z

Ω

*u(x)dθ*¯_{y}^{12}(x)

*−*Z

Ω

*u(x)dθ*¯^{21}* _{y}* (x) +
Z

Ω

*u(x)dθ*¯_{y}^{22}(x)

*dl(y)*

= 2 Z

Ω*×*Ω*0**u(x)δv(y)dθ*¯^{11}(x, y)*−*Z

Ω*×*Ω*0**u(x)δv(y)dθ*¯^{12}(x, y)

*−*Z

Ω*×*Ω^{0}

*u(x)δv(y)dθ*¯^{21}(x, y) +
Z

Ω*×*Ω^{0}

*u(x)δv(y)dθ*¯^{22}(x, y)

*,*

where *δv(y) :=* *v(y)−w*0(y) and we had in mind the disintegration formulae for *θ*^{11}, *θ*^{12}, *θ*^{21} and *θ*^{22} with
respect to*l.*

Let (θ_{y}^{11}) and (θ^{21}* _{y}* ) be the disintegrations of

*θ*

^{11}and

*θ*

^{21}with respect to

*ν*1, which is the projection of both on Ω

*. Similarly, let (θ*

^{0}^{12}

*) and (θ*

_{y}^{22}

*) be the disintegrations of*

_{y}*θ*

^{12}and

*θ*

^{22}with respect to

*ν*2. Then

*d*

*dtJ(θ** ^{t}*)

*|*

*t=0*= 2 Z

*B*_{1}

*δv(y)*
Z

*A*_{1}

1

*γu(x)dµ*1(x)*−*Z

*A*_{2}

*u(x)dθ*^{21}* _{y}* (x)

*dν*1(y)
+ 2

Z

*B*2

*δv(y)*
Z

*A*2

1

*γu(x)dµ*2(x)*−*Z

*A*2

*u(x)dθ*^{12}* _{y}* (x)

*dν*2(y).

Applying Property 4.3 in each point*y* *∈B*1 with the function *u*on*A*1 and*A*2, and the probability measures

1

*γ**µ*1 on*A*1 and *θ*^{21}* _{y}* on

*A*2, we obtain that

*α(y) :=*R

*A*_{1}
1

*γ**u(x)dµ*1(x)*−*R

*A*_{2}*u(x)dθ*_{y}^{21}(x)*<*0. Similar arguments
lead to*β*(y) :=R

*A*2

1

*γ**u(x)dµ*2(x)*−*R

*A*2*u(x)dθ*^{12}* _{y}* (x)

*>*0. With the above notations we have

*d*

*dtJ(θ** ^{t}*)

*|*

*= 2 Z*

^{t=0}*B*1

*δv(y)α(y)dν*1(y) +
Z

*B*2

*δv(y)β*(y)dν2(y)

*.* (4.1)

Note that the measures*−α(y)dν*1 and*β(y)dν*2 have the same mass. Indeed

*−*Z

*B*_{1}

*α(y)dν*1(y) =
Z

*B*_{1}

Z

*A*_{2}

*u(x)dθ*^{21}* _{y}* (x)dν1(y)

*−*Z

*B*_{1}

Z

*A*_{1}

1

*γu(x)dµ*1(x)dν1(y)

= Z

*A*2*×**B*1

*u(x)dθ*^{21}(x, y)*−*Z

*A*1*×**B*1

*u(x)dθ*^{11}(x, y) =
Z

*A*2

*u(x)dµ*2(x)*−*Z

*A*1

*u(x)dµ*1(x)

and

Z

*B*_{2}

*β(y)dν*2(y) =
Z

*A*_{2}*×**B*_{2}

1

*γu(x)dµ*2(x)dν2(y)*−*
Z

*A*_{1}*×**B*_{2}

*u(x)dθ*^{12}(x, y)

= Z

*A*_{2}

*u(x)dµ*2(x)*−*Z

*A*_{1}

*u(x)dµ*1(x).

Now we can apply Property 4.3 with the function *δv* on *B*1 and *B*2, and with the measures *−α(y)dν*1 on*B*1

and*β(y)dν*2 on*B*2, respectively, and from (4.1) we obtain that
*d*

*dtJ*(θ* ^{t}*)

*|*

^{t=0}*<*0 which contradicts the fact that

*θ*

^{0}is a minimum for

*J*.

**Step 3.**

Suppose that*v(y)−w*0(y)*6*= 0 on a set of positive*l*measure. We show then that there exists a real number*r*

that satisfies: Z

Ω*0*(v(y)*−r)*+*dl(y)>*

Z

Ω

(u(x)*−r)*+*dm(x),*

which is in contradiction with the hypothesis since Φ(λ) := (λ*−r)*+ is a continuous convex function (by*f*+we
represent the positive part of the function*f*). The number*r*is constructed from the following considerations:

Define for all*p∈*Rthe set*C**p* :=*{x∈*Ω*|u(x)> p} × {y* *∈*Ω^{0}*|v(y)−w*0(y)*<*0*}*. Note that if for some
*p*1 *∈* R, *θ*^{0}(C*p*1) = 0 then *θ*^{0}(C*p*) = 0 for every *p > p*1, since*C**p* *⊂* *C**p*1. Let *p*0 := inf*{p∈*R *|θ*^{0}(C*p*) = 0*}*.
Then*θ*^{0}(C*p*0) = 0 since*C**p*0 may be written as*C**p*0 :=S

*n**C**p**n* for some sequence*p**n**&p*0 and having in mind
that *θ*^{0}(C*p**n*) = 0 for each *n. For allp < p*0, *θ*^{0}(C*p*)*>*0 and by Step 2 it turns out that *θ*^{0}(*{x∈*Ω*|* *u(x)≤*
*p} × {y∈*Ω^{0}*|v(y)−w*0(y)*≥*0*}*) = 0. Taking a sequence*p**n**%p*0we get by employing the same arguments as
above, that*θ*^{0}(*{x∈*Ω*|u(x)< p*0*, v(y)−w*0(y)*≥*0*}*) = 0.

Analogously, define for all *q* *∈* R the set *D**q* := *{x* *∈* Ω *|* *u(x)* *< q} × {y* *∈* Ω^{0}*|* *v(y)−w*0(y) *>* 0*}*
and note that if for some *q*1 *∈* R, *θ*^{0}(D*q*1) = 0, then *θ*^{0}(D*q*) = 0 for every *q < q*1 since *D**q* *⊂* *D**q*1. Let
*q*0:= sup*{q∈*R*|θ*^{0}(D*q*) = 0*}*. By similar arguments to the ones used above with*C**p*0,*θ*^{0}(D*q*0) = 0. Moreover,
using a sequence*q**n**&q*0we obtain also that*θ*^{0}(*{x∈*Ω*|u(x)> q*0*} × {y∈*Ω^{0}*|v(y)−w*0(y)*≤*0*}*) = 0.

Note that *p*0 *≤q*0 otherwise, consider *s* such that*q*0 *< s < p*0. Then *θ*^{0}(C*s*)*>*0 and *θ*^{0}(D*s*)*>*0 that is,
*θ*^{0}(*{x∈*Ω*|u(x)> s} × {y∈*Ω^{0}*|v(y)−w*0(y)*<*0*}*)*>*0 and*θ*^{0}(*{x∈*Ω*|u(x)< s} × {y∈*Ω^{0}*|v(y)−w*0(y)*>*

0*}*)*>*0 which contradicts Step 2.

Consider now*r*=*q*0(any number between *p*0 and*q*0may be taken). Let us evaluate
Z

Ω

(u(x)*−q*0)+*dm(x) =*
Z

Ω*×*Ω*0*(u(x)*−q*0)+*dθ*^{0}(x, y) =
Z

*{x|u>q*0}×

*{y|v−w*0*>0}*

(u(x)*−q*0)dθ^{0}(x, y), (4.2)
where for the last equality we had in mind the above deduced relation*θ*^{0}(*{x∈* Ω *|* *u(x)* *> q*0*} × {y* *∈* Ω^{0}*|*
*v(y)−w*0(y)*≤*0*}*) = 0.

In order to evaluateR

Ω* ^{0}*(v(y)

*−q*0)+

*dl(y) let us first make the following analysis:*

In the set *{y* *∈* Ω^{0}*|* *v(y)−w*0(y) *<* 0*}*, since *θ*^{0}(C*p*_{0}) = 0 *i.e.* *θ*^{0}(*{x∈*Ω*|u(x)> p*0*} × {y∈*Ω^{0}*|v(y)*

*−w*0(y)*<*0*}*) = 0, and having in mind the definition *w*0 :=R

Ω*u(x)dθ*^{0}* _{y}*(x), it turns out that

*w*0(y)

*≤p*0 and therefore

*v(y)< w*0(y)

*≤p*0

*≤q*0.

In the set *{y* *∈* Ω^{0}*|* *v(y)−w*0(y) = 0*}*, since *θ*^{0}(*{x* *∈* Ω *|* *u(x)* *< p*0*} × {y* *∈* Ω^{0}*|* *v(y) =* *w*0(y)*}*) = 0
and *θ*^{0}(*{x* *∈*Ω *|* *u(x)* *> q*0*} × {y* *∈* Ω^{0}*|* *v(y) =* *w*0(y)*}*) = 0 as subsets of*{x∈* Ω *|* *u(x)< p*0*} × {y* *∈*Ω^{0}*|*
*v(y)−w*0(y)*≥*0*}*and respectively,*{x∈*Ω*|u(x)> q*0*} × {y∈*Ω^{0}*|v(y)−w*0(y)*≤*0*}*, we have that*w*0*∈*[p0*, q*0]
and in particular*v(y)≤q*0.

In the set*{y∈*Ω^{0}*|v(y)−w*0(y)*>*0*}*, since*θ*^{0}(D*q*0) = 0 we obtain that*w*0(y)*≥q*0 and hence*v(y)> q*0.
So the only points where (v*−q*0)+ does not vanish are*{y* *∈*Ω^{0}*|v(y)−w*0(y)*>*0*}*and in this set we have
*v(y)> w*0(y)*≥q*0.

Now we can evaluateR

Ω*0*(v(y)*−q*0)+*dl(y) as follows, where the first inequality occurs applying Property 4.3*
with the functions*v*and *w*0 on Ω* ^{0}* and with the same measure

*l:*

Z

Ω^{0}

(v(y)*−q*0)+*dl(y) =*
Z

*{**y**|**v**−**w*0*>0**}*(v(y)*−q*0)dl(y)*>*

Z

*{**y**|**v**−**w*0*>0**}*(w0(y)*−q*0)dl(y)

= Z

*{**y**|**v**−**w*_{0}*>0**}*

Z

Ω

*u(x)dθ**y*^{0}(x)*−q*0

*dl(y)*

= Z

Ω*×{**y**|**v**−**w*0*>0**}*(u(x)*−q*0)dθ^{0}(x, y)*≥*Z

*{x|u>q*0}×

*{y|v−w*0*>0}*

(u(x)*−q*0)dθ^{0}(x, y). (4.3)

Then (4.2) and (4.3) yield the contradiction.

The following lemma is the most important ingredient in the proof of Theorem 3.1. Tartar suggested the statement below and the main idea of its proof as well.

The domain*ω*2in the sequel is as in Theorem 3.1.

**Lemma 4.5.** *Given a sequencev**n*:*ω*2*→*[α, β], assume that it gives rise to the Young measure *µ*= (µ*y*). One
*defines a function* *g*:*ω*2*×*(0,1)*→*[α, β]*such that* *g(y,·*)*is nonincreasing and its distribution measure is* *µ**y**.*
*Then, given* *θ∈*[0,1], for all sequences*χ**n*:*ω*2*→ {*0,1*}such thatχ**n*** θ, up to a subsequence, we have that*

weak lim

*n* *χ**n**v**n* *≤*
Z *θ*

0

*g(y, s)ds.* (4.4)

*Moreover, there exists a sequenceχ*¯*n*:*ω*2*→ {*0,1*},χ*¯*n*** θ* *such that, up to a subsequence*

¯
*χ**n**v**n****

Z *θ*
0

*g(y, s)ds.* (4.5)

*Proof.* Consider a real number *θ* *∈* [0,1] arbitrarily fixed and consider a sequence of characteristic functions
*χ**n* ** θ. Denote by* *π* the Young measure associated to the pair (v*n**, χ**n*). So *π* = (π*y*)*y**∈**ω*_{2} is a measure
on [α, β]*× {*0,1*}*, where *π**y* = *π*_{y}^{1}*⊗δ**χ=0*+*π*_{y}^{2}*⊗δ**χ=1*. The projection of *π**y* on *{*0,1*}* is (1*−θ)δ*0+*θδ*1 and
the projection of *π**y* on [α, β] is *µ**y*. Therefore *µ**y* = *π*_{y}^{1}+*π*_{y}^{2} and since *π*_{y}^{1} and *π*_{y}^{2} are positive measures, it
turns out that*π*^{2}*y* is absolutely continuous with respect to*µ**y*. Then by Radon-Nikodym theorem we have that
*dπ*^{2}* _{y}*(v) =

*η*

*y*

*dµ*

*y*(v) where

*η*

*y*is a positive function in

*L*

^{1}

_{µ}*([α, β]) such that 0*

_{y}*≤η*

*y*

*≤*1

*µ*

*y*-almost everywhere in [α, β].

The weak limit of*χ**n* is*θ*so*π*_{y}^{2}([α, β]) =*θ. The weak limit ofχ**n**v**n* is calculated as follows:

*χ**n**v**n****
Z

*χvdπ**y*(χ, v) =
Z

*χvd(π*_{y}^{1}*⊗δ**χ=0*) +
Z

*χvd(π*_{y}^{2}*⊗δ**χ=1*) =
Z

*vdπ*_{y}^{2}(v) =
Z *θ*

0

*h(y, s)ds,* (4.6)
where *h* : *ω*2*×*(0, θ)*→* [α, β] is a nonincreasing function whose distribution measure is *π*^{2}* _{y}*. Consider

*h, for*instance, given by

*h(y, s) := sup{x∈*[α, β] :*π*^{2}* _{y}*(]x, β])

*> s}·*

Since for*g* one can take the following similar definition

*g(y, s) := sup{x∈*[α, β] :*µ**y*(]x, β])*> s},*

having in mind that*π*^{2}_{y}*≤µ**y*, one obtains that for all*y* *∈ω*2, *h(y, s)≤g(y, s), forL*^{1} almost every*s∈*]0, θ[.

ThusR*θ*

0 *h(y, s)ds≤*R*θ*

0 *g(y, s)ds*and then (4.6) implies (4.4).

We prove in the sequel that the equality is reached, up to a subsequence, that is, there exists a sequence

¯

*χ**k**→θ*such that ¯*χ**k**v**n*_{k}*→*R*θ*

0 *g(y, s)ds, for a subsequence (v**n** _{k}*)

*k*of (v

*n*)

*n*. Note that there exists

*c∈*[α, β] such that

*µ*

*y*(]c, β])

*≤θ*

*≤µ*

*y*([c, β]). Indeed, having in mind the definition of

*g, it is sufficient to take*

*c*=

*g(y, θ).*

Let us define

¯
*η**y*(v) =

1, if*v > c,*
*γ,* if*v*=*c,*
0, if*v < c,*
where*γ* = *θ−µ**y*(]c, β])

*µ**y*(*{c}*) if*µ**y*(*{c}*)*6*= 0 and*γ* may be any number between 0 and 1 if *µ**y*(*{c}*) = 0. Then the
following expression writes: Z

*vη*¯*y*(v)dµ*y*(v) =
Z

]c,β]

*vdµ**y*(v) +*cγµ**y*(*{c}*).

Denote by*I*the first term and by*II* the second term in the above sum. Let us calculate the*L*^{1}-measure of the
set *{s*:*g(y, s)> c}*:

*L*^{1}(*{s*:*g(y, s)> c}*) =
Z 1

0

*χ*_{{}*g(y,s)>c**}*(s)ds=
Z

[α,β]

*χ*_{{}*v>c**}*(v)dµ*y*(v) =*µ**y*(]c, β]).

Since*g(y,·*) is nonincreasing it turns out that*{s*:*g(y, s)> c}*is the interval with extremities 0 and*µ**y*(]c, β]).

Hence for all *s∈*]µ*y*(]c, β]), θ], we have that*g(y, s) =c* and then the first term*I* yields
*I*=

Z

]c,β]

*vdµ**y*(v) =
Z 1

0

*g(y, s)χ*_{{}*s:g(y,s)>c**}*(s)ds=

Z *µ** _{y}*(]c,β])
0

*g(y, s)ds.*

If*µ**y*(*{c}*)*6*= 0, the second term gives:

*II*=*c(θ−µ**y*(]c, β])) =*c*
Z *θ*

*µ** _{y}*(]c,β)

*ds*=
Z *θ*

*µ** _{y}*(]c,β])

*g(y, s)ds,*
and consequently

*I*+*II* =
Z *θ*

0

*g(y, s)ds.*

If*µ**y*(*{c}*) = 0 then*µ**y*(]c, β]) =*θ*=*µ**y*([c, β]) and*II* = 0 while
*I*=

Z

]c,β]

*vdµ**y*(v) =

Z *µ** _{y}*(]c,β])
0

*g(y, s)ds*=
Z *θ*

0

*g(y, s)ds.*

It remains now to show that there exists a sequence ¯*χ**n* such that the pair (v*n**,χ*¯*n*) gives rise to the Young
measure *π**y* =*π*_{y}^{1}*⊗δ*0+*π*^{2}_{y}*⊗δ*1 with*dπ*^{2}* _{y}* = ¯

*η*

*y*

*dµ*

*y*and

*dπ*

^{1}

*= (1*

_{y}*−η*¯

*y*)dµ

*y*. This existence is ensured by the following lemma applied with

*K*=

*{*0,1

*}*.

Let Ω be a bounded open set inR* ^{N}*.

**Lemma 4.6.** *Given* *v**n* *a sequence* *v**n* : Ω *→*[α, β] *giving rise to the Young measure* *µ* *and given a family of*
*probability measures*(π*x,v*)(x,v)*∈*Ω*×*[α,β] *with support inK(K* *is a bounded subset of*R^{p}*), there exists a sequence*

(χ*n*),*χ**n*: Ω*→K* *such that for all continuous functionsF* : [α, β]*×K→*R*we have, up to a subsequence:*

*F*(v*n**, χ**n*)***
Z

[α,β]*×**K*

*F(v, u)dπ**x,v*(u)dµ*x*(v),
*that is the sequence*(v*n**, χ**n*)*gives rise to the Young measureπ.*

The above result generalizes Theorem 5 in [9] (p. 147), and the proof uses analogous arguments.

*Proof.* Consider the following two sets:

*M*1=*{π*measure on Ω*×*[α, β]*×K* such that it is a narrow limit of Young measures
associated to (v*n**, χ) whereχ*: Ω*→K* is some measurable function*}*

and

*M*2=*{π*measure on Ω*×*[α, β]*×K, π≥*0,suppπ*⊂*Ω*×*[α, β]*×K,*proj_{Ω}_{×}_{[α,β]}*π*=*µ}·*

We prove that equality*M*1=*M*2 holds.

First step. We prove that*M*1 is convex.

Consider*π*1*, π*2*, . . . , π**q**∈M*1. Then each*π**i*is the narrow limit of Young measures associated to pairs (v*n**, χ**i*)
that is, for all bounded Carath´eodory integrands*φ,*

Z

[α,β]*×**K*

*φ(x, v, λ)dπ**i*(v, λ) = lim

*m* *φ(x, v**m*(x), χ*i*(x)).

We shall show that *π* :=P*q*

*i=1**π**i**θ**i* belongs to*M*1, where *θ**i* are real nonnegative numbers such thatP*q*
*i=1**θ**i*

= 1. By Theorem 3 in [9], there exist the following sequences of characteristic functions *ψ**i,n* : Ω *→ {*0,1*}*
such that P*q*

*i=1**ψ**i,n* = 1 and *ψ**i,n* ** θ**i*. Consider *χ**n* := P*q*

*i=1**ψ**i,n*(x)χ*i*(x). Then *φ(x, v**m*(x), χ*n*(x)) =
P*q*

*i=1**φ(x, v**m*(x), χ*i*(x))ψ*i,n*(x) and passing to the limit first in *m* and then in *n, one obtains that, for all*
bounded Carath´eodory integrands*φ,*

lim*n* lim

*m* *φ(x, v**m*(x), χ*n*(x)) =
Z

[α,β]*×**K*

*φ(x, v, λ)dπ(v, λ)*
and consequently*π∈M*1.

Second step. We prove that conv(M1) = *M*2. We know that in order to find the closed convex hull of*M*1

we need only to consider the affine continuous functions which are positive on*M*1(by Hahn-Banach theorem).

But an affine continuous function on the space of measures on Ω*×*[α, β]*×K*has the following form
*π→ hπ,*Φ0(x, v, λ)*i*+*δ*

where Φ0is continuous bounded and*δ∈*R. Such affine continuous function is positive on*M*1 if
*hπ,*Φ0(x, v, λ)*i*+*δ≥*0

for all*π∈M*1. This is equivalent to
lim*m*

Z

Ω

Φ0(x, v*m*(x), χ(x))dx+*δ≥*0 (4.7)

for all*χ*: Ω*→K. Define Ψ*0(x, v) := inf_{λ}_{∈}_{K}*{*Φ0(x, v, λ)*}*. Then Ψ0 is continuous bounded and therefore (4.7)
is equivalent to

lim*m*

Z

Ω

Ψ0(x, v*m*(x))dx+*δ≥*0.

Let Φ0(x, v, λ) = Ψ0(x, v) +*χ*0(x, v, λ), with*χ*0(x, v, λ)*≥*0. So, we have that

conv(M1) =*{π*measure that satisfies*hπ,*Φ0*i*+*δ≥*0 for all Φ0and*δ*satisfying
Φ0= Ψ0+*χ*0*,* for some function*χ*0*≥*0, and*h*Ψ0*, µi*+*δ≥*0*} ·*
Now it remains to prove that the above conditions characterize*M*2.

1) Take Ψ0= 0 and*δ*= 0. Then*hπ, χ*0*i ≥*0 for all*χ*0*≥*0. Therefore*π≥*0.

2) If*χ*0 = 0 in Ω*×*[α, β]*×K* then *χ*0 *≥*0 and *−χ*0 *≥*0. Then *hπ, χ*0*i ≥*0 and *hπ,−χ*0*i ≥* 0 and hence
*hπ, χ*0*i*= 0, therefore suppπ*⊂*Ω*×*[α, β]*×K.*

3) Consider *δ*=*−h*Ψ0*, µi*and *χ*0 = 0. Then*hπ,*Ψ0*i*+*δ* *≥*0 for all continuous bounded function Ψ0 such
that*δ*=*−h*Ψ0*, µi*. So,*hπ,*Ψ0*i ≥ hµ,*Ψ0*i*. On the other hand*hπ,−*Ψ0*i −δ≥*0. Then*hπ,*Ψ0*i ≥ hµ,*Ψ0*i*.
Hence*hπ,*Φ0*i*=*hµ,*Ψ0*i*, and consequently projΩ*×*[α,β]*π*=*µ.*

From 1), 2) and 3) above it turns out that conv(M1)*⊂M*2.

Conversely, let us prove that*M*2 *⊂* conv(M1). Consider *π* *∈* *M*2. Then *hπ,*Φ0*i*+*δ* *≥* 0 since *hπ,*Φ0*i* =
*hπ, χ*0*i*+*hπ,*Ψ0*i*,*hπ, χ*0*i ≥*0 and*hπ,*Ψ0*i*=*hµ,*Ψ0*i ≥*0. Hence*π∈*conv(M1).

Thus we obtain the existence of a sequence ¯*χ**m*: Ω*→K*that satisfies
weak lim

*m* weak lim

*n* *φ(x, v**n*(x),*χ*¯*m*(x)) =
Z

[α,β]*×**K*

*φ(x, v, λ)dπ**x*(v, λ),

for all bounded Carath´eodory integrands*φ. One can extract then a diagonal subsequencen**k* such that:

weak lim

*k* *φ(x, v**n** _{k}*(x),

*χ*¯

*n*

*(x)) = Z*

_{k}[α,β]*×**K*

*φ(x, v, λ)dπ**x*(v, λ).

I warmly thank Professor Guy Bouchitt´e for suggesting the study of problem (1.1). I am very grateful to Professor Luc Tartar, who conjectured the main result of this paper, for many stimulating conversations we have had together. This re- search work was supported by FEDER-PRAXIS/2/2.1/MAT/125/94, PRAXIS-FEDER/3/3.1/CTM/10/94, Universidade de Lisboa, JNICT-MENESR 97/166.

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