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Nonlocal Variational Principles with Variable Growth
This paper is concerned with the functional defined by , where is a regular open bounded set and 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 .
In this paper, we are concerned with a type of variational integrals that can be written as where is a regular open bounded set (), , and is Lipschitz continuous with . satisfies that and are measurable and and are continuous. is also assumed to satisfy the variable growth condition for . Notice that After Kováčik and Rákosník first discussed and spaces in , 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 -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 -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. When is a constant function, Pedregal studied problem (1) in . In connection with variational problems of nonlocal nature the reader can consult  for problems related to ferromagnetism,  about the regularization of a nonconvex problem, and [15–17] in order to analyze mechanical problems formulated in general context of nonlocal elasticity. In  some interesting tools to obtain a full relaxation of specific nonlocal variational problems have been analyzed, and  is also a remarkable work for a general class of nonlocal integral functionals. If convexity condition does not hold for , 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 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 Section 2, 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).
Let be the set of all Lebesgue measurable functions , where is a nonempty open subset. Denote where . The variable exponent Lebesgue space is the class of all functions such that for some . is a Banach space endowed with the norm (5). Equation (4) is called the modular of in .
For a given , we define the conjugate function as
Lemma 1 (see ). Let , then the inequality holds for every , .
In the following parts of this section, for every , we assume .
Lemma 2 (see ). For any , we have(1)if , then ;(2)if , then .
Lemma 3 (see ). If , is reflexive, and the dual space of is .
Lemma 4 (see ). Let , where denotes the Lebesgue measure of , and ; then the necessary and sufficient condition for is that for almost every , and in this case the embedding is continuous.
Next is a given positive integer. Given a multi-index , we set and , where is the generalized derivative operator.
The variable exponent Sobolev space is the class of functions on such that for every multi-index with . is a Banach space endowed with the norm By we denote the subspace of which is the closure of with respect to the norm (8).
For any , define Then is an equivalent norm of . If is a bounded domain, is an equivalent norm of .
Lemma 5 (see ). The spaces and are separable. Furthermore they are reflexive if .
Denote the dual space of by ; then we have the following.
Lemma 6 (see ). Let . Then for every , there exists such that
The norm of is defined as
Lemma 7 (see ). Let be a domain in with cone property. If is Lipschitz continuous and , is measurable and satisfies for almost every ; then there is a continuous embedding .
Lemma 8 (see ). Let be a domain in with cone property. If is continuous and , then for any measurable function defined in with for almost every and , there is a continuous compact embedding .
Lemma 9 (see ). Let ; then the Hardy-Littlewood maximal operator is bounded on .
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 ). 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 ). Assume that the sequence is bounded in . Then there exist a subsequence and a Borel probability measure on for a.e. , such that for each we have where We call the family of Young measure associated with the subsequence .
Lemma 11 (see ). Let be Lebesgue measurable (not necessarily bounded) and , , be a sequence of Lebesgue measurable functions. Then there exists a subsequence still denoted as and a family of nonnegative Radon measures on , such that(i) for almost every ;(ii) weakly* in for any , where and ;(iii)if for any Then for almost every , and for any measurable there holds weakly in for continuous provided that the sequence is weakly relative compact in .
Lemma 11 is the fundamental theorem of Young measure. A family satisfying (i)-(ii) always exists, and 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 13. It was shown in  that, in order to identify the Young measure associated with a particular sequence of functions , it is enough to check for every . It is even enough to have for and belonging to dense, countable subsets of and , respectively.
Lemma 11 has useful applications in nonlinear partial differential equation theory. The following lemmas are useful for us.
Lemma 14 (see ). Let be a uniformly bounded sequence in . Then there exist a subsequence (not relabeled), a nonincreasing sequence of measurable subsets of with as , and such that for all .
Lemma 15 (see ). Let be a sequence of vector-valued functions with associated Young measure . If, for , a nonnegative Carathéodory function, we have then for any measurable subset and for any in the space
Lemma 16 (see ). If is a sequence of measurable functions with associated parameterized measure , for every nonnegtive Carathéodory function and every measurable subset .
Lemma 17 (see ). If the sequence is bounded in , then there is a Young measure generated by satisfying , and the weak -limit of is .
Lemma 18 (see ). Let . If in , then the sequence generates the Young measure . Moreover, for almost every , is a probability measure and satisfies .
By Lemmas 3.8 and 3.9 in , we know that if a sequence is bounded in , then can generate a family of Young measures. Now we can define -Young measure for a sequence of functions in .
Definition 19. is a family of probability measures supported in . is called a -Young measure if it can be generated by gradients of a bounded sequence of functions in .
Theorem 20. Assume that the sequences and are bounded in . We have(a)if as , then the Young measures for both sequences are the same;(b)if as , then the Young measures for both sequences are the same.
Proof. (a) By Remark 13, it suffices to prove that the limits of and are the same for every and every . We conclude the result from
(b) For any and , we have Since , as , there exists a subsequence still denoted as such that , for almost every , as . Then , for almost every , as . By dominated convergence theorem, we get
Example 21. Assume that is a bounded sequence in and let be its associated Young measure. Consider the truncation operators By Lemma 2, there exists a constant such that . Then for any subsequence as , Thus the Young measure corresponding to is also .
Theorem 22. Let be a bounded sequence in such that the sequence generates the Young measure . Then we have(a)there exists such that ;(b)there exists a new sequence bounded in such that generates the same Young measure and for all . Moreover if is equi-integrable, so is .
Proof. (a) Since is a bounded sequence in , by Lemma 5, there exists a subsequence still denoted as such that in for some . Moreover in . By Lemmas 17 and 18, in . Thus .
(b) Let be a sequence of cut-off functions satisfying(i) on ;(ii) in ;(iii) for some constant .
Consider the sequence Clearly for all , , and Notice that Lemma 8 implies that as . Then we can choose a subsequence so that is bounded in and equi-integrable if is equi-integrable. Let be the sequence . Moreover Thus by Theorem 20, and generate the same Young measure.
Theorem 23. Let be a bounded sequence in . Then there always exists another sequence of Lipschitz functions with for all such that is equi-integrable and the two sequences of gradients, and , generate the same -Young measure.
Proof. By Lemma 6, we can assume that in for some . By Theorem 22, we can assume that . Extend by 0 to all of . By density, we can find such that as . This implies that is a bounded sequence in . Consider the sequence , where and is the Hardy-Littlewood maximal operator. By Lemma 9, is bounded in . Let be the corresponding Young measure (possibly for an appropriate subsequence). For fixed , is bounded in and for any , We can get by Dunford-Pettis and Lemma 11 By Levei theorem, we have and . Moreover Thus we can get And we can find a subsequence as such that By Lemma 15, let . For any , there exists a constant such that . We can take . Thus That is, in . Let . Since is bounded in and as , . There exists Lipschitz function , such that in and for all . Because , by Theorem 20 again, and generate the same Young measure. Since , we have It follows from the equi-integrability of that is equi-integrable. Take . Then the sequence verifies the conclusion of the theorem.
Theorem 24. Let be a family of probability measures supported in . is the Young measure generated by a sequence , where is a bounded sequence in . Then where is the Young measure generated by the sequence of gradients .
Proof. Let and . We have
Remark 25. Concerning the above result it must be pointed out that we have the representation for any continuous such that converges weakly in .
4. Weak Lower Semicontinuity and Relaxation
Theorem 26. Assume that (2) holds. is weakly lower semicontinuous in if and only if for any -Young measure ,
Proof. For any -Young measure , by Definition 19, can be generated by a bounded sequence in . Moreover we can assume that in (possibly for a suitable subsequence). By Lemma 18, we have . According to Theorem 22, there exists another sequence such that is weakly convergent in . and generate the same Young measure. Going if necessary to a subsequence, we can assume that is also weakly convergent to . Actually if , by Lemma 18, we have . Then we can take , and is also weakly convergent in . Since is weakly lower semicontinuous and (2) holds, we can get On the other hand, if in , then by Lemma 18 can generate a family of -Young measure and . Thus (2) and Lemma 16 imply that
Theorem 27. Let in and be the -Young measure associated to (or possibly to a subsequence). Then for all measurable , if and only if
Proof . In view of Theorem 26, we only need show the necessity. By Lemma 14, there exists a sequence of measurable sets such that for each , , and
in for all . By (47) and Lemma 11
Lemma 16 implies that and by (47) . Therefore
Then we can get the conclusion.
In many models from mathematical physics we need to consider variational integrals like where is some fixed function in . The energy of a Young measure is defined by where is a -Young measure generated by a sequence of gradients in ; there exists , such that , . Let
Theorem 28. Let Then and is indeed a minimum.