# Nonlocal Variational Principles with Variable Growth

Authors: Yongqiang Fu; Miaomiao Yang
Journal: Journal of Function Spaces
 (2014)

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## Abstract

This paper is concerned with the functionalJ defined by J(u)=∫Ω×ΩW(x,y,∇u(x),∇u(y))dxdy, where Ω⊂ℝN is a regular open bounded set and W is a real-valued function with variable growth. After discussing the theory of Young measures in variable exponent Sobolev spaces, we study the weak lower semicontinuity and relaxation of J.


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## Body

## 1. Introduction

In this paper, we are concerned with a type of variational integrals that can be written as(1)J(u)=∫Ω×Ω‍W(x,y,∇u(x),∇u(y))dxdy,
where Ω⊂ℝN is a regular open bounded set (N≥1), u∈W1,p(x)(Ω;ℝm), and p(x) is Lipschitz continuous with 1<p-:=inf⁡x∈Ω¯p(x)≤p(x)≤p+:=sup⁡x∈Ω¯p(x)<∞. W=W(x,y,λ1,λ2):ℝN×ℝN×𝕄m×N×𝕄m×N→ℝ satisfies that x↦W(x,y,λ1,λ2) and y↦W(x,y,λ1,λ2) are measurable and λ1↦W(x,y,λ1,λ2) and λ2↦W(x,y,λ1,λ2) are continuous. W is also assumed to satisfy the variable growth condition
(2)c(|λ1|p(x)+|λ2|p(x)-1)≤W(x,y,λ1,λ2)≤C(|λ1|p(x)+|λ2|p(x)+1),
for 0<c<C. Notice that
(3)Lp(x)(Ω,ℝm)={u:Ω⊂ℝN⟶ℝm,|u|∈Lp(x)(Ω,ℝ)},W1,p(x)(Ω,ℝm)={u:Ω⊂ℝN⟶ℝm,u∈Lp(x)(Ω,ℝm),∇u∈Lp(x)(Ω,𝕄m×N)}.
After Kováčik and Rákosník first discussed Lp(x)(Ω) and Wm,p(x)(Ω) spaces in [1], a lot of research has been done concerning these kinds of variable exponent spaces; for examples, see [2–6] for the properties of such spaces and [7–9] for the applications of variable exponent spaces on partial differential equations. These problems with variable exponent growth possess very complicated nonlinearities; for instance, the p(x)-Laplacian operator is inhomogeneous. In recent years, these problems have received considerable attention and raised many difficult mathematical problems. The theory on various mathematical problems with p(x)-growth conditions has important applications in nonlinear elastic mechanics, imaging processing, electrorheological fluids. and other physics phenomena (see [10, 11]).The main goal of this paper is to study the weak lower semicontinuity and relaxation of the nonlocal variational problems. We will analyze these problems in terms of Young measures generated by sequences in variable exponent space. Whenp(x) is a constant function, Pedregal studied problem (1) in [12]. In connection with variational problems of nonlocal nature the reader can consult [13] for problems related to ferromagnetism, [14] about the regularization of a nonconvex problem, and [15–17] in order to analyze mechanical problems formulated in general context of nonlocal elasticity. In [18] some interesting tools to obtain a full relaxation of specific nonlocal variational problems have been analyzed, and [19] is also a remarkable work for a general class of nonlocal integral functionals. If convexity condition does not hold for W, then we would like to have an equivalent variational principle in the sense that the infimum is preserved and the relaxed variational principle admits a minimizer. The usual way to proceed for local functionals is to replace W by its convex hull. In nonlocal setting, there is no substitute for the convex hull. The only possible way to describe relaxation for these nonlocal variational principles is by using Young measures and a generalized functional defined on them (see [12, 20]).This paper is organized as follows. In Section2, several important properties on variable exponent spaces are recalled; in Section 3, we give some conclusions of Young measures in variable exponent spaces; in Section 4, we analyze the weak lower semicontinuity and relaxation of problem (1).


## 2. Preliminaries

In this section, we first recall some facts on variable exponent spacesLp(x)(Ω) and Wk,p(x)(Ω). See [1, 4–6] for the details.LetP(Ω) be the set of all Lebesgue measurable functions p:Ω→[1,+∞], where Ω⊂ℝn(n≥2) is a nonempty open subset. Denote
(4)ρp(x)(u)=∫Ω∖Ω∞‍|u(x)|p(x)dx+sup⁡Ω∞|u(x)|,(5)∥u∥p(x)=inf⁡{t>0:ρp(x)(ut)≤1},
where Ω∞={x∈Ω:p(x)=∞}. The variable exponent Lebesgue space Lp(x)(Ω) is the class of all functions u such that ρp(x)(t0u)<∞ for some t0>0. Lp(x)(Ω) is a Banach space endowed with the norm (5). Equation (4) is called the modular of u in Lp(x)(Ω).For a givenp(x)∈P(Ω), we define the conjugate function p′(x) as
(6)p′(x)={∞,ifx∈Ω1={x∈Ω:p(x)=1},1,ifx∈Ω∞,p(x)p(x)-1,forotherx∈Ω.Lemma 1 (see [6]).
Letp∈P(Ω), then the inequality
(7)∫Ω‍|u(x)v(x)|dx≤2∥u∥p(x)∥v∥p′(x)
holds for every u∈Lp(x)(Ω), v∈Lp′(x)(Ω).In the following parts of this section, for everyp∈P(Ω), we assume 1≤p-≤p(x)≤p+<∞.Lemma 2 (see [5]).
For anyu∈Lp(x)(Ω), we have(1)
if∥u∥p(x)≥1, then ∥u∥p(x)p-≤ρp(x)(u)≤∥u∥p(x)p+;(2)
if∥u∥p(x)<1, then ∥u∥p(x)p+≤ρp(x)(u)≤∥u∥p(x)p-.Lemma 3 (see [5]).
Ifp->1, Lp(x)(Ω) is reflexive, and the dual space of Lp(x)(Ω) is Lp′(x)(Ω).Lemma 4 (see [1]).
Let|Ω|<∞, where |Ω| denotes the Lebesgue measure of Ω, and p1(x),p2(x)∈P(Ω); then the necessary and sufficient condition for Lp2(x)(Ω)⊂Lp1(x)(Ω) is that p1(x)≤p2(x) for almost every x∈Ω, and in this case the embedding is continuous.Nextk is a given positive integer. Given a multi-index α=(α1,…,αn)∈ℕn, we set |α|=α1+⋯+αn and Dα=D1α1⋯Dnαn, where Di=∂/∂xi is the generalized derivative operator.The variable exponent Sobolev spaceWk,p(x)(Ω) is the class of functions u on Ω such that Dαu∈Lp(x)(Ω) for every multi-index α with |α|≤k. Wk,p(x)(Ω) is a Banach space endowed with the norm
(8)∥u∥k,p(x)=∑|α|≤k‍∥Dαu∥p(x).
By W0k,p(x)(Ω) we denote the subspace of Wk,p(x)(Ω) which is the closure of C0∞(Ω) with respect to the norm (8).For anyu∈W1,p(x)(Ω), define
(9)∥|u|∥=inf⁡{t>0:∫Ω‍|∇u|p(x)+|u|p(x)tp(x)dx≤1}.
Then ∥|·|∥ is an equivalent norm of W1,p(x)(Ω). If Ω is a bounded domain, ∥∇u∥p(x) is an equivalent norm of W01,p(x)(Ω).Lemma 5 (see [1]).
The spacesWk,p(x)(Ω) and W0k,p(x)(Ω) are separable. Furthermore they are reflexive if p->1.Denote the dual space ofW0k,p(x)(Ω) by W-k,p′(x)(Ω); then we have the following.Lemma 6 (see [1]).
Letp∈P(Ω)∩L∞(Ω). Then for every G∈W-k,p′(x)(Ω), there exists gα∈{gα∈Lp′(x)(Ω):|α|≤k} such that
(10)G(u)=∑|α|≤k‍∫Ω‍Dαu(x)gα(x)dx,∀u∈W0k,p(x)(Ω).The norm ofW-k,p′(x)(Ω) is defined as
(11)∥G∥-k,p′(x)=sup⁡{|G(u)|∥u∥k,p(x):u∈Wk,p(x)(Ω)∖{0}}.Lemma 7 (see [4]).
LetΩ be a domain in ℝn with cone property. If p:Ω¯→ℝ is Lipschitz continuous and 1<p-≤p+<n/k, q:Ω¯→ℝ is measurable and satisfies p(x)≤q(x)≤p*(x):=np(x)/(n-kp(x)) for almost every x∈Ω¯; then there is a continuous embedding Wk,p(x)(Ω)↪Lq(x)(Ω).Lemma 8 (see [4]).
LetΩ be a domain in ℝn with cone property. If p:Ω¯→ℝ is continuous and 1<p-≤p+<n/k, then for any measurable function q(x) defined in Ω with p(x)≤q(x) for almost every x∈Ω¯ and inf⁡x∈Ω¯(p*(x)-q(x))>0, there is a continuous compact embedding Wk,p(x)(Ω)↪↪Lq(x)(Ω).Lemma 9 (see [6]).
Letp∈P(Ω); then the Hardy-Littlewood maximal operator is bounded on Lp(x)(Ω).


## 3. Young Measure Generated by Sequences in Variable Exponent Space

Weak convergence is a basic tool of modern nonlinear analysis because it has the same compactness properties as the convergence in finite dimensional spaces (see [21]). But this notion does not behave as we desire with respect to nonlinear functionals and operators. Young measures are a device to overcome these difficulties. For the details we refer to [21–24]. Inspired by these works, we will discuss Young measures in variable exponent spaces. In some references Young measures are called parameterized measures. Here we take the two terms as equivalent. First, we recall the definition and some Lemmas on Young measures.Definition 10 (see [21]).
Assume that the sequence{fk}k=1∞ is bounded in L∞(Ω;ℝm). Then there exist a subsequence {fkj}j=1∞⊂{fk}k=1∞ and a Borel probability measure νx on ℝm for a.e. x∈Ω, such that for each F∈C(ℝm) we have
(12)F(fkj)⇀*F¯weakly*inL∞(Ω),
where
(13)F¯(x)≡∫ℝm‍F(y)dνx(y)(a.e.x∈Ω).
We call {νx}x∈U the family of Young measure associated with the subsequence {fkj}j=1∞.Lemma 11 (see [22]).
LetΩ⊂ℝN be Lebesgue measurable (not necessarily bounded) and zk:Ω→ℝm, k=1,2,…, be a sequence of Lebesgue measurable functions. Then there exists a subsequence still denoted as {zk} and a family {νx}x∈Ω of nonnegative Radon measures on ℝn, such that(i)
∥νx∥:=∫‍dνx≤1 for almost every x∈Ω;(ii)
φ(zk)⇀*φ¯ weakly* in L∞(Ω) for any φ∈C0(ℝm), where φ¯(x)=〈νx,φ〉 and C0(ℝm)={φ∈C(ℝm):lim⁡|z|→∞|φ(z)|=0};(iii)
if for anyR>0(14)lim⁡L→∞sup⁡k∈ℕmeas⁡{x∈Ω∩B(0,R):|zk(x)|≥L}=0.Then ∥νx∥=1 for almost every x∈Ω, and for any measurable A⊂Ω there holds φ(zk)⇀φ¯=〈νx,φ〉 weakly in L1(A) for continuous φ provided that the sequence φ(zk) is weakly relative compact in L1(A).Lemma11 is the fundamental theorem of Young measure. A family {νx}x∈Ω satisfying (i)-(ii) always exists, and νx is a Young measure if (14) holds. The proof and complete analysis of Lemma 11 can be found in [20, 23]. According to the comments there, we give some remarks.Remark 12.
Ball showed in [23] that under hypothesis (14) for any measurable A⊂Ω(15)f(x,zk(x))⇀〈νx,f(x,·)〉=∫ℝm‍f(x,λ)dνx(λ)inL1(A),
for every Caratheodory function f:A×ℝm→ℝ such that {f(x,zk(x))} is sequentially weakly relative compact in L1(A).Remark 13.
It was shown in [20] that, in order to identify the Young measure associated with a particular sequence of functions {zj}, it is enough to check
(16)φ(zj)⇀*φ¯inL∞(Ω),
for every φ∈C0(ℝm). It is even enough to have
(17)lim⁡j→∞∫Ω‍ξ(x)φ(zj(x))dx=∫Ω‍ξ(x)∫ℝm‍φ(λ)dνx(λ)dx,
for ξ and φ belonging to dense, countable subsets of L1(Ω) and C0(ℝm), respectively.Lemma11 has useful applications in nonlinear partial differential equation theory. The following lemmas are useful for us.Lemma 14 (see [20]).
Let{fj} be a uniformly bounded sequence in L1(Ω). Then there exist a subsequence (not relabeled), a nonincreasing sequence of measurable subsets Ωn of Ω with meas⁡Ωn→0 as n→∞, and f∈L1(Ω) such that
(18)fj⇀finL1(Ω∖Ωn),
for all n.Lemma 15 (see [20]).
Let{zj} be a sequence of vector-valued functions with associated Young measure ν={νx}x∈Ω. If, for φ0, a nonnegative Carathéodory function, we have
(19)lim⁡j→∞∫Ω‍φ0(x,zj(x))dx=∫Ω‍∫ℝm‍φ0(x,λ)dνx(λ)dx<∞,
then
(20)lim⁡j→∞∫E‍φ(x,zj(x))dx=∫E‍∫ℝm‍φ0(x,λ)dνx(λ)dx<∞,
for any measurable subset E⊂Ω and for any φ in the space
(21)ℰφ0={φ:φisaCarathe´odoryfunctionwith|φ|≤C(1+φ0)}.Lemma 16 (see [20]).
If{zi} is a sequence of measurable functions with associated parameterized measure ν={νx}x∈Ω,
(22)liminf⁡j→∞∫E‍ψ(x,zj(x))dx≥∫E‍∫ℝm‍ψ(x,λ)dνx(λ)dx,
for every nonnegtive Carathéodory function ψ and every measurable subset E⊂Ω.Lemma 17 (see [25]).
If the sequence{uj} is bounded in Lp(x)(Ω,ℝm), then there is a Young measure νx generated by {uj} satisfying ∥νx∥=1, and the weak L1-limit of {uj} is ∫ℝm‍λdνx(λ).Lemma 18 (see [25]).
Letmeas⁡Ω<∞. If uk⇀u in W01,p(x)(Ω,ℝm), then the sequence {(uk,Duk)} generates the Young measure δu(x)⊗νx. Moreover, for almost every x∈Ω, νx is a probability measure and satisfies 〈νx,I〉=Du(x).By Lemmas 3.8 and  3.9 in [25], we know that if a sequence {uj} is bounded in W1,p(x)(Ω,ℝm), then {∇uj} can generate a family of Young measures. Now we can define W1,p(x)-Young measure for a sequence of functions in W1,p(x)(Ω,ℝm).Definition 19.
{
ν
x
}
x
∈
Ω is a family of probability measures supported in 𝕄m×N. {νx}x∈Ω is called a W1,p(x)-Young measure if it can be generated by gradients of a bounded sequence of functions in W1,p(x)(Ω,ℝm).Theorem 20.
Assume that the sequences{zj} and {wj} are bounded in Lp(x)(Ω,ℝm). We have(a)
ifmeas⁡{zj≠wj}→0 as j→∞, then the Young measures for both sequences are the same;(b)
if∥zj-wj∥Lp(x)(Ω,ℝm)→0 as j→∞, then the Young measures for both sequences are the same.Proof.
(a) By Remark13, it suffices to prove that the limits of ∫Ω‍ξ(x)φ(zj(x))dx and ∫Ω‍ξ(x)φ(wj(x))dx are the same for every ξ∈L1(Ω) and every φ∈C0(ℝm). We conclude the result from
(23)|∫Ω‍ξ(x)φ(zj(x))dx-∫Ω‍ξ(x)φ(wj(x))dx|=|∫{zj≠wj}‍ξ(x)φ(zj(x))dx-∫{zj≠wj}‍ξ(x)φ(wj(x))dx|≤|∫{zj≠wj}‍ξ(x)φ(zj(x))dx|+|∫{zj≠wj}‍ξ(x)φ(wj(x))dx|≤∫{zj≠wj}‍2∥φ∥L∞(Ω)|ξ(x)|dx⟶0,
as j→∞.
(b) For anyξ∈L1(Ω) and φ∈C0(ℝm), we have
(24)|ξ(x)(φ(zj(x))-φ(wj(x)))|≤2∥φ∥L∞(Ω)|ξ(x)|.
Since ∥zj-wj∥Lp(x)(Ω,ℝm)→0, as j→∞, there exists a subsequence still denoted as {zj-wj} such that zj-wj→0, for almost every x∈Ω, as j→∞. Then φ(zj(x))-φ(wj(x))→0, for almost every x∈Ω, as j→∞. By dominated convergence theorem, we get
(25)lim⁡j→∞∫Ω‍ξ(x)φ(zj(x))dx=lim⁡j→∞∫Ω‍ξ(x)φ(wj(x))dx.Example 21.
Assume that{zj} is a bounded sequence in Lp(x)(Ω,ℝm) and let ν={νx}x∈Ω be its associated Young measure. Consider the truncation operators
(26)Tk(s)={s,|s|<k,ks|s|,|s|≥k.
By Lemma 2, there exists a constant C such that ρp(x)(zj)≤C. Then for any subsequence k(j)→∞ as j→∞,
(27)|{|zj|≥k(j)}|≤Ck(j)p-⟶0.
Thus the Young measure corresponding to Tk(j)(zj) is also ν.Theorem 22.
Let{vj} be a bounded sequence in W1,p(x)(Ω,ℝm) such that the sequence {∇vj} generates the Young measure ν={νx}x∈Ω. Then we have(a)
there existsu∈W1,p(x)(Ω,ℝm) such that ∇u(x)=∫𝕄m×N‍λdνx(λ)∈𝕄m×N;(b)
there exists a new sequence{uk} bounded in W1,p(x)(Ω,ℝm) such that {∇uk} generates the same Young measure ν and uk-u∈W01,p(x)(Ω,ℝm) for all k. Moreover if {|∇vj|p(x)} is equi-integrable, so is {|∇uk|p(x)}.Proof.
(a) Since{vj} is a bounded sequence in W1,p(x)(Ω,ℝm), by Lemma 5, there exists a subsequence still denoted as {vj} such that vj⇀u in W1,p(x)(Ω,ℝm) for some u∈W1,p(x)(Ω,ℝm). Moreover ∇vj⇀∇u in Lp(x)(Ω,𝕄m×N)⊂L1(Ω,𝕄m×N). By Lemmas 17 and 18, ∇vj⇀∫𝕄m×N‍λdνx(λ) in L1(Ω,𝕄m×N). Thus ∇u(x)=∫𝕄m×N‍λdνx(λ).
(b) Let{ηk} be a sequence of cut-off functions satisfying(i)
ηk=1 on ∂Ω;(ii)
ηk=0 in Ωk={x∈Ω:dist⁡(x,∂Ω)≥1/k};(iii)
|∇ηk|≤Ck for some constant C.
Consider the sequence{wjk}(28)wjk(x)=ηk(x)u(x)+(1-ηk(x))vj(x).
Clearly wjk-u∈W01,p(x)(Ω,ℝm) for all j, k, and
(29)∇wjk(x)=ηk(x)∇u(x)+(1-ηk(x))∇vj(x)+(u(x)-vj(x))·∇ηk(x).
Notice that
(30)∫Ω‍|ηk(x)∇u(x)|p(x)dx≤∫Ω‍|∇u(x)|p(x)dx,∫Ω‍|(1-ηk(x))∇vj(x)|p(x)dx≤∫Ω‍|∇vj(x)|p(x)dx,∫Ω‍|u(x)-vj(x)|p(x)|∇ηk(x)|p(x)dx≤max⁡{(Ck)p-,(Ck)p+}∫Ω‍|u(x)-vj(x)|p(x)dx.
Lemma 8 implies that ∥vj-u∥Lp(x)(Ω,ℝm)→0 as j→0. Then we can choose a subsequence {j(k)} so that {|∇wj(k),k|p(x)} is bounded in L1(Ω) and equi-integrable if {|∇vj|p(x)} is equi-integrable. Let {uk} be the sequence {wj(k),k}. Moreover
(31)|{∇vj(x)≠∇wj(k),k}|=|{ηk>0}|⟶0,ask⟶∞.
Thus by Theorem 20, {∇uk} and {∇vj} generate the same Young measure.Theorem 23.
Let{vj} be a bounded sequence in W1,p(x)(Ω,ℝm). Then there always exists another sequence {uj} of Lipschitz functions with uj∈W1,∞(Ω,ℝm) for all j such that {|∇uj|p(x)} is equi-integrable and the two sequences of gradients, {∇vj} and {∇uj}, generate the same W1,p(x)-Young measure.Proof.
By Lemma6, we can assume that vj⇀u in W1,p(x)(Ω,ℝm) for some u∈W1,p(x)(Ω,ℝm). By Theorem 22, we can assume that vj-u∈W01,p(x)(Ω,ℝm). Extend wj=vj-u by 0 to all of ℝN. By density, we can find zj∈C0∞(ℝN,ℝm) such that ∥zj-wj∥W1,p(x)(ℝN,ℝm)→0 as j→∞. This implies that {zj} is a bounded sequence in W1,p(x)(Ω,ℝm). Consider the sequence {M*(zj)}, where
(32)M*(zj)=M(|zj|)+M(|∇zj|)
and M is the Hardy-Littlewood maximal operator. By Lemma 9, {M*(zj)} is bounded in Lp(x)(ℝN). Let μ={μx}x∈ℝN be the corresponding Young measure (possibly for an appropriate subsequence). For fixed k, {TkM*(zj)} is bounded in L∞(ℝN) and for any E⊂ℝN,
(33)∫E‍|TkM*(zj)|p(x)dx≤∫E‍kp(x)dx≤kp+meas⁡E.
We can get by Dunford-Pettis and Lemma 11(34)lim⁡j→∞∫ℝN‍|TkM*(zj)|p(x)dx=∫ℝN‍∫ℝ‍|Tk(x)|p(x)dμx(s)dx.
By Levei theorem, we have
(35)lim⁡k→∞∫ℝ‍|Tk(s)|p(x)dμx(s)=∫ℝ‍lim⁡k→∞|Tk(s)|p(x)dμx(s)=∫ℝ‍|s|p(x)dμx(s):=g(x)
and g(x)∈L1(ℝN). Moreover
(36)lim⁡k→∞∫ℝN‍∫ℝ‍|Tk(s)|p(x)dμx(s)dx=∫ℝN‍g(x)dx.
Thus we can get
(37)lim⁡k→∞lim⁡j→∞∫ℝN‍|TkM*(zj)|p(x)dx=lim⁡k→∞∫ℝN‍∫ℝ‍|Tk(s)|p(x)dμx(s)dx=∫ℝN‍g(x)dx.
And we can find a subsequence k(j)→∞ as j→∞ such that
(38)lim⁡j→∞∫ℝN‍|Tk(j)M*(zj)|p(x)dx=∫ℝN‍g(x)dx.
By Lemma 15, let φ0(x,λ)=|λ|p(x). For any ξ∈L∞(ℝN), there exists a constant C such that |ξ(x)|≤C. We can take φ(x,λ)=ξ(x)|λ|p(x)≤C(1+φ0). Thus
(39)∫ℝN‍ξ(x)|Tk(j)M*(zj)|p(x)dx⟶∫ℝN‍∫ℝ‍ξ(x)|s|p(x)dμx(s)dx=∫ℝN‍ξ(x)g(x)dx.
That is, |Tk(j)M*(zj)|p(x)⇀g in L1(ℝN). Let Aj={M*(zj)≥k(j)}. Since {M*(zj)} is bounded in Lp(x)(ℝN) and k(j)→∞ as j→∞, |Aj|→0. There exists Lipschitz function u¯j, such that u¯j=zj in ℝN∖Aj and |∇u¯j|≤C(N)k(j) for all j. Because {∇u¯j≠∇zj}⊂Aj, by Theorem 20 again, {∇u¯j} and {∇zj} generate the same Young measure. Since M*(zj)≥|∇zj|, we have
(40)|∇u¯j|p(x)≤max⁡{C(N)p-,C(N)p+}|Tk(j)M*(zj)|p(x).
It follows from the equi-integrability of {|Tk(j)M*(zj)|p(x)} that {|∇u¯j|p(x)} is equi-integrable. Take uj=u¯j|Ω+u. Then the sequence {uj} verifies the conclusion of the theorem.Theorem 24.
LetΠ={Π(x,y)} be a family of probability measures supported in 𝕄m×N×𝕄m×N. Π is the Young measure generated by a sequence {gj(x,y)}={(∇uj(x),∇uj(y))}, where {uj} is a bounded sequence in W1,p(x)(Ω,ℝm). Then
(41)Π(x,y)=νx⊗νy,(x,y)∈Ω×Ω,
where ν={νx}x∈Ω is the Young measure generated by the sequence of gradients {∇uj}.Proof.
Letφ=φ1φ2 and θ=θ1θ2. We have
(42)∫Ω×Ω‍θ1(x)θ2(y)×(∫𝕄m×N×𝕄m×N‍φ1(λ1)φ2(λ2)dΠ(x,y)(λ1,λ2))dxdy=lim⁡j→∞∫Ω×Ω‍θ1(x)θ2(y)φ(gj(x,y))dxdy=lim⁡j→∞∫Ω×Ω‍θ1(x)θ2(y)φ1(∇uj(x))×φ2(∇uj(y))dxdy=lim⁡j→∞(∫Ω‍θ1(x)φ1(∇uj(x))dx)×(∫Ω‍θ2(y)φ2(∇uj(y))dy)=∫Ω‍θ1(x)(∫𝕄m×N‍φ1(λ1)dνx(λ1))dx×∫Ω‍θ2(x)(∫𝕄m×N‍φ2(λ2)dνy(λ2))dy=∫Ω×Ω‍θ1(x)θ2(y)×∫𝕄m×N×𝕄m×N‍φ1(λ1)φ2(λ2)d(νx⊗νy)(λ1,λ2)dxdy.Remark 25.
Concerning the above result it must be pointed out that we have the representation(43)lim⁡j→∞∫Ω×Ω‍ψ(∇uj(x),∇uj(y))dxdy=∫Ω×Ω‍∫𝕄m×N×𝕄m×N‍ψ(λ1,λ2)dνx(λ1)dνy(λ2)dxdy
for any continuous ψ such that {ψ(∇uj(x),∇uj(y))}j converges weakly in L1(Ω×Ω).


## 4. Weak Lower Semicontinuity and Relaxation

Theorem 26.
Assume that (2) holds. J is weakly lower semicontinuous in W1,p(x)(Ω,ℝm) if and only if for any W1,p(x)-Young measure ν={νx}x∈Ω,
(44)∬ΩW(x,y,∫𝕄m×N‍λ1dνx(λ1),∫𝕄m×N‍λ2dνy(λ2))dxdy≤∬Ω∬𝕄m×NW(x,y,λ1,λ2)dνx(λ1)dνy(λ2)dxdy.Proof.
For anyW1,p(x)-Young measure ν, by Definition 19, ν can be generated by a bounded sequence {uj} in W1,p(x)(Ω,ℝm). Moreover we can assume that uj⇀u in W1,p(x)(Ω,ℝm) (possibly for a suitable subsequence). By Lemma 18, we have ∇u(x)=∫𝕄m×N‍λdνx(λ). According to Theorem 22, there exists another sequence {u¯j} such that {|∇u¯j|p(x)} is weakly convergent in L1(Ω). {∇u¯j} and {∇uj} generate the same Young measure. Going if necessary to a subsequence, we can assume that {u¯j} is also weakly convergent to u. Actually if u¯j⇀u¯≠u, by Lemma 18, we have ∇u¯=∫𝕄m×N‍λdνx(λ)=∇u. Then we can take u¯j-u¯+u⇀u, and {|∇(wj-u¯+u)|p(x)} is also weakly convergent in L1(Ω). Since J(u) is weakly lower semicontinuous and (2) holds, we can get
(45)∬ΩW(x,y,∫𝕄m×N‍λ1dνx(λ1),∫𝕄m×N‍λ2dνy(λ2))dxdy=∬ΩW(x,y,∇u(x),∇u(y))dxdy≤lim⁡j→∞∬ΩW(x,y,u¯j(x),u¯j(y))dxdy=∬Ω∬𝕄m×NW(x,y,λ1,λ2)dνx(λ1)dνy(λ2)dxdy.
On the other hand, if uj⇀u in W1,p(x)(Ω,ℝm), then by Lemma 18  {uj} can generate a family of W1,p(x)-Young measure ν={νx}x∈Ω and ∇u(x)=∫𝕄m×N‍λdνx(λ). Thus (2) and Lemma 16 imply that
(46)lim⁡j→∞∬ΩW(x,y,∇uj(x),∇uj(y))dxdy≥∬Ω∬𝕄m×NW(x,y,λ1,λ2)dνx(λ1)dνy(λ2)dxdy≥∬ΩW(x,y,∫𝕄m×N‍λ1dνx(λ1),∫𝕄m×N‍λ2dνy(λ2))dxdy=∬ΩW(x,y,∇u(x),∇u(y))dxdy.Theorem 27.
Letuj⇀u in W1,p(x)(Ω,ℝm) and ν={νx}x∈Ω be the W1,p(x)-Young measure associated to {∇uj} (or possibly to a subsequence). Then
(47)∬EW(x,y,∇u(x),∇u(y))dxdy≤lim⁡⁡inf⁡j→∞∬EW(x,y,∇uj(x),∇uj(y))dxdy,
for all measurable E⊂Ω, if and only if
(48)W(x,y,∇u(x),∇u(y))≤∬𝕄m×NW(x,y,λ1,λ2)dνx(λ1)dνy(λ2),a.e.(x,y)∈Ω×Ω.Proof .
In view of Theorem26, we only need show the necessity. By Lemma 14, there exists a sequence of measurable sets {Ωk} such that Ωk+1⊂Ωk for each k, lim⁡k→∞|Ωk|=0, and
(49)W(x,y,∇uj(x),∇uj(y))⇀∬𝕄m×NW(x,y,λ1,λ2)dνx(λ1)dνy(λ2)
in L1((Ω∖Ωk)×(Ω∖Ωk)) for all k. By (47) and Lemma 11(50)∬E∖ΩkW(x,y,∇u(x),∇u(y))dxdy≤lim⁡j→∞∬E∖ΩkW(x,y,∇uj(x),∇uj(y))dxdy=∬E∖Ωk∬𝕄m×NW(x,y,λ1,λ2)dνx(λ1)dνy(λ2)dxdy.
Lemma 16 implies that ∫𝕄m×N‍∫𝕄m×N‍W(x,y,λ1,λ2)dνx(λ1)dνy(λ2)∈L1(Ω×Ω) and by (47) W(x,y,∇u(x),∇u(y))∈L1(Ω×Ω). Therefore
(51)lim⁡k→∞∬E∖Ωk∬𝕄m×NW(x,y,λ1,λ2)dνx(λ1)dνy(λ2)dxdy=0,lim⁡k→∞∬E∖ΩkW(x,y,∇u(x),∇u(y))dxdy=0.
Moreover
(52)∬EW(∇u(x),∇u(y))dxdy≤∬E∬𝕄m×NW(x,y,λ1,λ2)dνx(λ1)dνy(λ2)dxdy.
Then we can get the conclusion.
In many models from mathematical physics we need to consider variational integrals like(53)J(u)=∬ΩW(x,y,∇u(x),∇u(y))dxdy,u∈W1,p(x)(Ω,ℝm),u-u0∈W01,p(x)(Ω,ℝm),
where u0 is some fixed function in W1,p(x)(Ω,ℝm). The energy of a Young measure is defined by
(54)J¯(μ)=∬𝕄m×NW(x,y,λ1,λ2)dμx(λ1)dμy(λ2),
where μ is a W1,p(x)-Young measure generated by a sequence of gradients in W1,p(x)(Ω,ℝm); there exists u∈W1,p(x)(Ω,ℝm), such that ∇u(x)=∫𝕄m×N‍λdμx(λ), u-u0∈W01,p(x)(Ω,ℝm). Let
(55)𝒜={u∈W1,p(x)(Ω,ℝm):u-u0∈W01,p(x)(Ω,ℝm)},𝒜¯={u-u0∈W01,p(x)(Ω,ℝm)μisaW1,p(x)-Youngmeasure:∃u∈W1,p(x)(Ω,ℝm),∇u(x)=∫𝕄m×N‍λdμx(λ),u-u0∈W01,p(x)(Ω,ℝm)}.Theorem 28.
Let(56)m=inf⁡{J(u):u∈𝒜},(57)m¯=inf⁡{J¯(μ),μ∈𝒜¯}.
Then m=m¯ and m¯ is indeed a minimum.Proof.
Consider the following steps.
Step  1. Equation (56) admits a minimizing sequence {uj} such that {|∇uj|p(x)} is weakly convergent in L1(Ω).
Let{vj} be a minimizing sequence for (56). By (2), {vj} is a bounded sequence in W1,p(x)(Ω,ℝm). By Lemma 17, let ν={νx}x∈Ω denote the Young measure associated to the sequence of gradients. According to Theorem 22, there exists another sequence {wj} such that {|∇wj|p(x)} is weakly convergent in L1(Ω) and {∇wj} and {∇vj} generate the same Young measure. Going if necessary to a subsequence, we can assume that both vj and wj are weakly convergent to u. If wj⇀w≠u, by Lemma 18, we have ∇w=∫𝕄m×N‍Adνx(A)=∇u. Then we can take wj-w+u⇀u, and {|∇(wj-w+u)|p(x)} is also weakly convergent in L1(Ω). By Theorem 22, there exists a sequence {uj} in W1,p(x)(Ω,ℝm) such that uj-u∈W01,p(x)(Ω,ℝm) for every j. Moreover {∇wj} and {∇uj} generate the same Young measure and {|∇uj|p(x)} is also equi-integrable. Since vj∈𝒜 and 𝒜 is weakly closed, we can get u∈𝒜. Moreover uj∈𝒜. Since vj is a minimizing sequence for (56), we can get, by Lemma 16,
(58)∫Ω×Ω‍∫𝕄m×N×𝕄m×N‍W(x,y,λ1,λ2)dμx(λ1)dμy(λ2)dxdy≤lim⁡j→∞J(vj)≤lim⁡j→∞J(uj)=∫Ω×Ω‍∫𝕄m×N×𝕄m×N‍W(x,y,λ1,λ2)dμx(λ1)dμy(λ2)dxdy.
Then we can conclude that {uj} is also a minimizing sequence to (56).
Step  2
(
m
=
m
¯
)
. From Step 1, we know that ν∈𝒜¯. And
(59)m=lim⁡j→∞J(uj)=lim⁡j→∞∫Ω×Ω‍W(x,y,∇uj(x),∇uj(y))dxdy=∫Ω×Ω‍∫𝕄m×N×𝕄m×N‍W(x,y,λ1,λ2)dνx(λ1)dνy(λ2)dxdy≥m¯.
On the other hand, for anyμ∈𝒜¯, there exists a bounded sequence {uj} in W1,p(x)(Ω,ℝm) such that uj∈𝒜 for every j. We can assume that uj⇀u and {|∇uj|p(x)} is equi-integrable. Thus
(60)m≤lim⁡j→∞J(uj)=∫Ω×Ω‍∫𝕄m×N×𝕄m×N‍W(x,y,λ1,λ2)dμx(λ1)dμy(λ2)dxdy.
The arbitrariness of μ implies that m≤m¯.
Step  3. There exists ν such that
(61)J¯(ν)=inf⁡J¯(μ).
Ifuj is a minimizing sequence for (56), we can assume that {|∇uj|p(x)} is equi-integrable and the associated Young measure ν∈𝒜. Hence
(62)m¯=m=lim⁡j→∞J(uj)=∫Ω×Ω‍∫𝕄m×N×𝕄m×N‍W(x,y,λ1,λ2)dνx(λ1)dνy(λ2)dxdy.
Consequently, ν is a minimizer of (57).



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*Source: 435247-2014-04-07.xml*