Convergence of solutions of bilateral problems in variable domains and related questions Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 3, No. 2, 2017

CONVERGENCE OF SOLUTIONS OF BILATERAL PROBLEMS IN VARIABLE DOMAINS AND RELATED QUESTIONS

Alexander A. Kovalevsky

Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences; Ural Federal University, Ekaterinburg, Russia alexkvl71@mail.ru

Abstract: We discuss some results on the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. We consider the case of regular constraints, i.e., constraints lying in the corresponding Sobolev space, and the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function. The first case concerns a larger class of integrands and requires the positivity almost everywhere of the difference between the upper and lower constraints. In the second case, this requirement is absent. Moreover, in the latter case, the exhaustion condition of an n-dimensional domain by a sequence of n-dimensional domains plays an important role. We give a series of results involving this condition. In particular, using the exhaustion condition, we prove a certain convergence of sets of functions defined by bilateral (generally irregular) constraints in variable domains.

Key words: Integral functional, Bilateral problem, Minimizer, Minimum value, r-convergence of functionals, Strong connectedness of spaces, H-convergence of sets, Exhaustion condition.

Introduction

This paper is mainly based on the talk given by the author at the International S.B. Stechkin Summer Workshop-Conference on Function Theory, Miass, Russia, August 1-10, 2017.

The problems considered in the paper are related to the following general problem. Let {Ws} be a sequence of Banach spaces, and let, for every s € N, Is : Ws ^ R and Vs C Ws, Vs = 0. Let, for every s € N, us be a minimizer of Is on Vs. The questions are, what are general conditions under which the sequence {us} converges in a certain sense to an element and this limit element minimizes a functional I on a set V, and how are the functional I and the set V related to the sequences {Is} and {Vs} ? Problems of this kind are studied in the framework of homogenization theory. There is a special kind of convergence of functionals that helps to solve the mentioned problems. This is the r-convergence. There are many works devoted to the study of this convergence. The r-convergence of functionals with the same domain of definition was studied, for instance, in [1-3]. In the simplest case, the definition of r-convergence is as follows.

Definition 1. Let, for every s € N, fs : R ^ R, and let f : R ^ R. We say that the sequence {fs} T-converges to the function f if the following conditions are satisfied:

(a) for every x € R, there exists a sequence {ys} C R such that ys ^ x and fs(ys) ^ f (x);

(b) for every x € R and every sequence {xs} C R such that xs ^ x, we have the inequality liminf fs(xs) ^ f(x).

s—^^o

The r-convergence of ordinary real functions and functionals defined on Banach spaces has some interesting properties that distinguish it from other kinds of convergence of the corresponding mappings. Among various properties of the r-convergence, we only mention its variational property that describes the relation of this convergence of functionals to the convergence of their minimizers

and minimum values. A simple version of the variational property of the r-convergence is the following proposition.

Proposition 1. Let, for every s € N, fs : R ^ R, and let f : R ^ R. Assume that the sequence {fs} r-converges to the function f. Let, for every s € N, xs be a minimizer of fs on R. Assume that Xs ^ X. Then x minimizes f on R and fs(xs) ^ f (x).

Proof. Since x s ^ x, by condition (b) in Definition 1, we have

liminf fs(xs) ^ f (x). (1)

s—^^o

Now, let y € R. By virtue of condition (a) in Definition 1, there exists a sequence {ys} C R such that

fs(ys) ^ f (y). (2)

Since, for every s € N, xs minimizes fs on R, we have

Vs € N, fs(xs) < fs(ys). (3)

Relations (2) and (3) imply that

limsup fs(xs) < f (y). (4)

From (1) and (4), we derive that x minimizes f on R and fs(xs) ^ f (x). We note that the latter limit relation follows from inequality (1) and from inequality (4) with y = x. □

Here, we have restricted ourselves only to a simplest version of the variational property of the r-convergence, having shown how both conditions (a) and (b) in Definition 1 work. The considered case is very simple not only due the fact that we dealt with functions defined on R but also because of the assumption that the minimizers of these functions are global. In the case of minimizers on sets defined by certain constraints, the situation is more complicated, and not always the "global" r-convergence (i.e., the convergence of the kind described in Definition 1 with a T-realizing sequence {ys} taken in the whole corresponding space) can be used for the study of the convergence of such minimizers.

There are analogues of the above definition of r-convergence for functionals defined on a Banach space (in particular, on a Lebesgue or Sobolev space). In this connection, see, for instance, [2, 4]. The notion of r-convergence of functionals with varying domain of definition (in particular, of functionals Is : Wm'p(Qs) ^ R with taking into account the structure of domains Qs) was introduced and studied, for instance, in [5-7].

Next, note that, in the study of the convergence of minimizers us of functionals Is : Ws ^ R, a connection of the spaces Ws with a space W plays an important role. Often, this connection is expressed as the requirement that there exists a sequence of operators 1s : Ws ^ W with certain properties. In particular, these properties should provide the following property: for every sequence vs € Ws such that sup ||vs||Ws < the sequence {1svs} is bounded in W. Under appro-

s

seN

priate and in some sense natural conditions on the functionals Is, for the sequence of minimizers us € Ws of the functionals Is, the inequality sup ||us||Ws < holds. Therefore, if there exists

seN

a sequence 1s : Ws ^ W with the above mentioned property, then the sequence {1sus} is bounded. Consequently, if the space W is reflexive, there exist an increasing sequence {sj} C N and an element u € W such that 1sjusj ^ u weakly in W. Actually, this is the first step in the study of the convergence of the sequence of minimizers us € Ws of the functionals Is. The described idea with the operators 1s is realized in the justification of the results stated below for functionals defined

on the Sobolev spaces W 1,p(Qs), where {Qs} is a sequence of domains contained in a bounded domain Q of Rn. Essentially, the mentioned idea goes back to [8]. In this connection, see also [5-7].

The main content of this paper is organized as follows. In Section 1, we state the initial assumptions and the necessary definitions. In Section 2, we present our results on the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. We consider the case of regular constraints, i.e., constraints lying in the corresponding Sobolev space (see [9]), and the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function (in this connection, see [10]). In both cases, a certain connection of the spaces W 1,p(Qs) with the space W 1,p(Q) and the r-convergence of functionals defined on the spaces W 1,p(Qs) to a functional defined on W 1,p(Q) are essentially used. At the same time, some other conditions on the involved domains, integrands, and constraints are also important for our convergence results. On the whole, the conditions providing these results are discussed in Section 3, where a special attention is paid to the so-called exhaustion condition of the domain Q by the domains Qs. This condition is the requirement that, for every increasing sequence {mj} C N, the measure of the union of all the domains Qmj is equal to the measure of the domain Q. We also consider the notion of H-convergence of sequences of sets Us C W 1,p(Qs) to a set U C W 1,p(Q) and show the importance of the exhaustion condition for the H-convergence of sets of functions defined by irregular bilateral constraints.

1. Assumptions and definitions

Let n € N, n ^ 2, let Q be a bounded domain of Rn, and let p > 1. Let {Qs} be a sequence of domains of Rn contained in Q.

It is easy to see that if v € W 1>p(Q) and s € N, then v|Qs € W 1,p(Qs).

Definition 2. If s € N, then qs : W 1,p(Q) — W 1,p(Qs) is the mapping such that, for every function v € W 1,p(Q), we have qsv = v|ns.

Definition 3. We say that the sequence of spaces W 1,p(Qs) is strongly connected with the space W 1,p(Q) if there exists a sequence of linear continuous operators 1s : W 1,p(Qs) — W 1,p(Q) such that:

(a) the sequence of norms ||1s|| is bounded;

(b) for every s € N and for every v € W 1,p(Qs), we have qs(1sv) = v a.e. in Qs.

The prototype of the notion in Definition 3 is the condition of strong connectedness of n-dimensional domains introduced in [8].

Definition 4. Let, for every s € N, Is : W 1,p(Qs) — R, and let I: W 1>p(Q) — R. We say that the sequence {Is} T-converges to the functional I if the following conditions are satisfied:

(a) for every function v € W 1,p(Q), there exists a sequence ws € W 1,p(Qs) such that ||ws - ?sv||lp(ns) — 0 and Is(ws) — I(v);

(b) for every function v € W 1,p(Q) and for every sequence vs € W 1,p(Qs) such that ||vs - qs v||lp (ns) — 0, we have lii—inf Is (vs) ^ I (v).

Next, let c1,c2 > 0, and let, for every s € N, € L1(Qs) and ^ 0 in Qs. We assume that the sequence of norms ||^s||Li(ns) is bounded.

Let, for every s € N, fs : Qs x Rn — R be a function satisfying the following conditions: for every £ € Rn, the function fs(-,£) is measurable on Qs; for almost every x € Qs, the function fs(x, ■) is convex on Rn; for almost every x € Qs and for every £ € Rn, we have

Ciieip - ^(x) < < C2ie ip+Mx).

(5)

In view of the assumptions on the functions fs and for every s € N and for every v € W 1,P(Qs), the function fs(x, Vv) is summable on Qs.

Definition 5. If s € N, then Fs : W1,P(Qs) ^ R is the functional such that, for every function v € W 1>p(Qs), we have

Fs(v) = y fs(x, Vv)dx. ns

By virtue of the conditions on the functions fs, for every s € N, the functional Fs is convex and locally bounded. Therefore, for every s € N, the functional Fs is weakly lower semicontinuous.

Let c3, c4 > 0, and let, for every s € N, Gs : W 1,P(Qs) ^ R be a weakly continuous functional. We assume that, for every s € N and for every v € W 1,P(Qs),

Gs(v) ^ CsIMILp(Qs) - C4• (6)

Obviously, for every s € N, the functional Fs + Gs is weakly lower semicontinuous. Moreover, in view of (5) and (6) and the boundedness of the sequence of norms ||^s||li(ns), there exist positive constants c5 and such that, for every s € N and for every v € W 1,P(Qs), we have

(Fs + Gs)(v) ^ C5||v||Wl,p(ns) - c6• (7)

Thus, in view of the known results on the existence of minimizers of functionals (see, for instance, [11]), if s € N and Us is a sequentially weakly closed set in W 1,P(Qs), then there exists a minimizer of the functional Fs + Gs on the set

2. Variational problems with bilateral constraints

First, we consider the case of regular bilateral constraints.

Let p, ^ € W 1,P(Q), and let p ^ ^ a.e. in Q. We define

V(p,^) = {v € W 1>P(Q) : p < v < ^ a.e. in Q},

and let, for every s € N,

Vs(p,^) = {v € W 1>P(Qs) : p < v < ^ a.e. in Qs}.

It is easy to see that the set V(p, is nonempty, closed, and convex. Similarly, for every s € N, the set Vs(p, is nonempty, closed, and convex.

Clearly, for every s € N, there exists a function belonging to the set Vs(p, and minimizing the functional Fs + Gs on this set.

Theorem 1. Assume that the following conditions are satisfied:

(*1) the embedding of W 1,P(Q) into LP(Q) is compact;

(*2) the sequence of spaces W 1iP(Qs) is strongly connected with the space W 1iP(Q);

(*3) for every sequence of measurable sets Hs c Qs such that meas Hs ^ 0, we have

J dx ^ 0;

Hs

(*4) the sequence {Fs} Y-converges to a functional F : W 1,P(Q) ^ R;

(*5) there exists a functional G : W 1,P(Q) ^ R such that, for every function v € W 1,P(Q) and for every sequence vs € W 1iP(Qs) with the property ||vs — qsv||Lp(qs) ^ 0, we have Gs(vs) ^ G(v);

(*6) 0 — ^ > 0 a.e. in Q.

Let, for every s € N, us be a function in Vs(0, 0) minimizing the functional Fs + Gs on the set Vs(^>,0). Then there exist an increasing sequence {sj} C N and a function u € V(^>,0) such that u minimizes the functional F + G on the set V(^>,0), ||usj — qsju||LP(qs ) — 0, and (Fsj + Gsj )(usj) — (F + G)(u). '

Essentially, a similar result was obtained in [12] but under stronger assumptions on the func-tionals Fs and Gs and under the condition 0 — <p ^ a a.e. in Q, where a > 0. In this connection, see also [13, Theorem 2.9].

Concerning the proof of Theorem 1, we note the following. First, using operators 1s : W 1,p(Qs) — W 1,p(Q) described in Definition 3 and defining the functions us = min{max{1sus, ^>},0}, we find that there exist an increasing sequence {sj} C N and a function u € W 1,p(Q) such that usj — u strongly in Lp(Q) and almost everywhere in Q. Then we obtain the inclusion u € V(^>,0), the limit relation ||usj — qsju||LP(qs ) — 0, and, by virtue of conditions (*4) and (*5) of Theorem 1, the inequality liminf(Fs. + Gs.)(us.) ^ (F + G)(u). The next and most important step is to establish, for every function v € V(^>,0), the existence of a sequence ws € Vs(^>,0) with the following properties: ||ws — qsv||LP(Qs) — 0 and

limsup Fs (ws) ^ F(v). (8)

The construction of such a sequence involves the function v and a T-realizing sequence {vs} for v, i.e., a sequence vs € W 1,p(Qs) such that ||vs — qsv||Lp(qs) — 0 and Fs(vs) — F(v), which exists in view of condition (*4) of Theorem 1. Moreover, it involves the difference 0 — Using the limit relation ||vs — qsv||Lp(qs) — 0 and condition (*6) of Theorem 1, we find that, for a sequence {cts} C (0,1] converging to 0, meas{|vs — qsv| ^ asqs(0 — <^)} — 0. This is a key moment in the proof of inequality (8). For further details leading to the required properties of the function u, see [9, Section 2].

We now proceed to the case of irregular bilateral constraints. More precisely, we consider the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function. Thus, in contrast to the previous case, the upper constraint can be irregular and both constraints can coincide on a set of positive measure. This is due to an additional condition on the domains Qs and a stronger condition on the functions as compared to condition (*3) of Theorem 1.

Let 0 : Q —> R and 0 ^ 0 a.e. in Q. We define

V(0) = {v € W 1>p(Q) : 0 ^ v ^ 0 a.e. in Q},

and let, for every s € N,

Vs(0) = {v € W 1>p(Qs) : 0 < v < 0 a.e. in Qs}.

It is easy to see that the set V(0) is nonempty, closed, and convex. Moreover, for every s € N, the set Vs(0) is nonempty, closed, and convex.

Obviously, for every s € N, there exists a function belonging to the set Vs(0) and minimizing the functional Fs + Gs on this set.

Theorem 2. Assume that conditions (*1), (*2), (*4), and (*5) of Theorem 1 are satisfied. In addition, suppose that the following conditions are satisfied:

(*') for every increasing sequence {mj} C N, we have meas( Q \ (J Qm. ) = 0;

V j=1 V

(*") II^||l1(qs) — 0;

Let, for every s € N, us be a function in Vs(0) minimizing the functional Fs + Gs on the set Vs(0). Then there exist an increasing sequence {sj} c N and a function u € V(0) such that u minimizes the functional F + G on the set V(0), ||usj — qsju||Lp(qs ,) — 0, and (Fsj + Gsj)(usj) — (F + G) (u). '

As for the proof of Theorem 2, we give the following remarks. Since, in general, the function 0 is irregular, we cannot use functions like the above functions us in the proof of Theorem 1. Therefore, using operators 1s : W 1,P(Qs) — W 1,P(Q) described in Definition 3, first, we find that there exist an increasing sequence {sj} c N and a function u € W 1,P(Q) such that 1Sjusj — u strongly in LP(Q) and almost everywhere in Q. Then, to prove that u € V(0), along with the inclusions us € Vs(0), we use condition (*') of Theorem 2 which effectively works in this situation. Similarly to the proof of Theorem 1, the most important step in the proof of Theorem 2 is to establish, for every function v € V(0), the existence of a sequence ws € Vs(0) such that ||ws — qsv||LP(Qs) — 0 and inequality (8) holds. The construction of such a sequence involves the function v and a r-realizing sequence {vs} for v but does not involve the constraint 0. To prove inequality (8), we essentially use condition (*'') of Theorem 2 and the fact that meas({|vs — qsv| ^ asqsv} n {v > 0}) — 0, where {cts} is a sequence in [0,1) such that — 0. For details, see the proof of Theorem 3.1 in [10].

The next result describes a situation where we have the convergence of the whole sequence of minimizers and of the whole sequence of minimum values.

Theorem 3. Assume that conditions (*1), (*2), (*4), and (*5) of Theorem 1 are satisfied, and the functional G is .strictly convex on the set V(0). In addition, suppose that conditions (*') and (*'') of Theorem 2 are satisfied. Let, for every s € N, us be a function in Vs(0) minimizing the functional Fs + Gs on the set Vs(0). Then there exists a unique function u € V(0) minimizing the functional F + G on the set V(0) and the following relations hold: ||us — qsu||LP(Qs) — 0 and (Fs + Gs)(us) — (F + G) (u). S

3. Comments to the conditions of Theorems 1—3

As is known (see, for instance, [14, Chapter 6]), condition (*1) of Theorem 1 is satisfied if Q is a Lipschitz domain. In particular, bounded convex domains are Lipschitz domains. A more general requirement guaranteeing the fulfillment of condition (*1) is that Q is an extension domain (see, for instance, [15, Chapter 1]).

Condition (*2) of Theorem 1 is satisfied, in particular, if the domains Qs have a certain perforated structure. In this regard, see, for instance, [16, Section 2].

As far as conditions (*3) and (*4) of Theorem 1 are concerned, we note the following. In the case where the functions take a constant value independent of s, theorems on conditions for the r-convergence of the integral functionals Fs with the integrands fs satisfying condition (5) follow from the results of [17, 18], where the r-convergence of integral functionals defined on the spaces Wm'P(Qs) with an arbitrary m € N was studied. In this case, the sequence {Fs} T-converges to an integral functional defined on the space W 1iP(Q), in particular, if the domains Qs have a periodic perforated structure and all the integrands fs coincide with the same integrand having a certain regularity (see [17]). Obviously, in the specified case for the functions the sequence of norms ||^s||l1(qs) is bounded and condition (*3) of Theorem 1 is satisfied. In the more general case where € L1^) and ^ 0 in Qs for every s € N and, in addition, the inequality

lim sup / dx ^ ^ dx (9)

J J

QnQs QnQ

holds for a function ^ € L1(Q), ^ ^ 0 in Q, and for every open cube Q of Rn, a theorem on the r-compactness of the sequence {Fs} can be proved similarly to the corresponding results in [19, 20]. Obviously, in this case, the sequence of norms ||^s||Li(ns) is bounded. We also note that there are examples of sequences of nonnegative functions € L1(Qs) for which condition (9) and condition (*3) of Theorem 1 are satisfied but there is no function : Q — R such that, for every s € N, ^ a.e. in Qs. Such examples can be given with the use of the functions constructed in [21].

In connection with condition (*5) of Theorem 1, we give the following example.

Example 1. Let a € Lp/(p-1)(Q). Let ft € (0,1), let ft > 0, and let $ : [0, — R be a continuous function such that

Vn € [0, |$(n)| < ft|n|p + (10)

For every s € N, we define the functional Gs : W 1,p(Qs) — R by

Gs(v) = y{|v|p + av}dx + $(||v||LP(Qs)), v € W 1p(Qs).

ns

In view of (10), for every s € N and for every v € W 1,p(Qs), inequality (6) holds with constants c3 and c4 depending only on p, ft, ft, and ||a||Lp/(p-i)(Q). We also note that if conditions (*1) and (*2) of Theorem 1 are satisfied, then, for every s € N, the functional Gs is weakly continuous. Next, assume that the following condition is satisfied:

(*) there exists a nonnegative bounded measurable function b : Q — R such that, for every open cube Q C Q, we have meas(Q n Qs) — /

Jq

Now, let G : W 1,p(Q) — R be the functional such that, for every function v € W 1,p(Q), we have

G(v) = J b{|v|p + av}dx + $(||b1/pv||LP(n)). (11)

n

Using condition (*) and the continuity of the function $, we find that, for the sequence of func-tionals Gs, condition (*5) of Theorem 1 is satisfied.

We remark that if the domain Q is Lipschitz and the domains Qs have a certain periodically perforated structure, then conditions (*1) and (*2) of Theorem 1 are satisfied along with condition (*) in which the function b takes a constant positive value. Obviously, for such a function b, the functional G defined by (11) is strictly convex if the function $ is nondecreasing and convex.

We emphasize the importance of condition (*6) of Theorem 1 for its conclusion. In [9], we gave an example where all the conditions of Theorem 1 are satisfied except for condition (*6) but the conclusion of this theorem does not hold on the whole. We note that, in this example, for an arbitrary pre-assigned positive e, the measure of the set where the lower and upper constraints coincide does not exceed e. Here is a simple example where condition (*6) of Theorem 1 is satisfied.

Example 2. Let Q = {x € Rn : |x| < 1}, and let, for every x € Q, we have <^(x) = 0 and

o

0(x) = |x|2(1 — |x|2). In view of these assumptions, we have 0 € W 1,p(Q) and ^ ^ 0 in Q. In addition, for every x € Q \ {0}, (0 — <^)(x) > 0. Thus, condition (*6) of Theorem 1 is satisfied. We

o

observe that, in the case considered here, we have V(<£>,0) = {v € W 1,p(Q) : ^ ^ v ^ 0 a.e. in Q}. Hence, for p = 2, the set V(<£>, 0) has the same form as the set defined by bilateral constraints

bdx. iQ

in [22]. We also note that if uj is a domain of Rra such that uJ c!i and the origin is contained in w, then there is no number > 0 such that 0 — p ^ a.e. in w. We remark in this connection that it was shown in [22] that the G-convergence of a sequence of linear continuous divergence operators

o o

As : W 1,2(Q) ^ W-1'2(Q) to an operator A : W 1,2(Q) ^ W-1'2(Q) of the same form implies the weak convergence of solutions of variational inequalities with the operators As and the set of

o

constraints K(01,02) = {v € W 1,2(Q) : 01 ^ v ^ 02 a.e. in Q} to the solution of the corresponding variational inequality with the operator A and the same set of constraints. At the same time, it was assumed in [22] that 0^02 € L2(Q) and, for every subdomain w CC Q, there exist a number

o

> 0 and functions 0^,0£ € W 1>2(Q) such that 01 ^ 0^ ^ 0£ ^ 02 in Q and 0£ — 0^ ^ in w. Obviously, the functions p and 0 defined at the beginning of this example do not satisfy the assumption given in [22].

We now discuss condition (*') of Theorem 2. This condition is essential for the conclusion of Theorem 2. In [10], we construct an example where all the conditions of Theorem 2 are satisfied except for condition (*') but the conclusion of this theorem does not hold. We call condition (*') of Theorem 2 the exhaustion condition of the domain Q by the domains Qs. This condition plays an important role in the study of the convergence of solutions of variational problems with irregular unilateral and bilateral constraints in variable domains. In this regard, in addition to the present paper, see [23, 24]. We used the same exhaustion condition earlier in [6] for the investigation of both a convergence of sets in variable Sobolev spaces and the coercivity of the T-limit of functionals defined on these spaces. Below, we show how such questions are solved for sequences of sets Us C W 1,p(Qs) and the functionals Fs + Gs. Before we do this, let us give some useful results.

Proposition 2. Condition (*') of Theorem 2 is equivalent to the following condition:

if v € L^Q) and liminf / |v|dx = 0, then v = 0 a.e. in Q. (12)

S^-TO J

ns

Proof. Assume that condition (*') of Theorem 2 is satisfied. Let v € L^Q), and let

liminf / |v|dx = 0. s^^ J

ns

Fixing an arbitrary e > 0, we find that there exists an increasing sequence {sj} C N such that

£

Vie N, J\v\dx^^j. (13)

Setting Q' = |J Qs., by condition (*') of Theorem 2, we have meas(Q \ Q') = 0. Then

j=1 j

J |v|dx = J |v|dx ^ J |v|dx.

This and (13) imply that

Q Q' j=1 Qsj

J |v|dx ^ e.

Q

Hence, in view of the arbitrariness of e, we conclude that v = 0 a.e. in Q. Thus, condition (12) is satisfied.

Conversely, assume that condition (12) is satisfied. Let {mj} be an increasing sequence in N.

Setting E0 = Q \ U Qmj, we suppose that meas E0 > 0. Let x : Q — R be the characteristic

j=1

function of the set E0. Obviously, x € L1(Q) and / xdx = 0 for every j € N. Therefore,

J Qmj

lim inf / x dx = 0.

Then, by condition (12), we have x = 0 a.e. in Q. Hence, there exists a set E c Q of measure zero such that, for every x € Q \ E, we have x(x) = 0. Then, fixing x € E0 \ E, we obtain x(x) = 0. On the other hand, by the definition of the function x, we have x(x) = 1. The obtained contradiction proves that meas E0 = 0. Thus, condition (*') of Theorem 2 is satisfied. □

Proposition 3. Let condition (*') of Theorem 2 be satisfied. Then the following condition is satisfied:

if v € W 1,p(Q) and lim inf ||qsv||Lp(Q ) = 0, then v = 0 a.e. in Q. (14)

Proof. Let v € W 1,p(Q) and lim inf ||qsv||Lp(Q ) = 0. Setting w = |v|p, we have

w € L1 (Q), lim inf / wdx = 0. (15)

s^^ J Qs

Since, by assumption, condition (*') of Theorem 2 is satisfied, we deduce from Proposition 2 that condition (12) is satisfied. The latter condition along with (15) implies that w = 0 a.e. in Q. Hence, v = 0 a.e. in Q. Thus, condition (14) is satisfied. □

Proposition 4. Let condition (*1) of Theorem 1 be satisfied, and assume that there exists a sequence of linear continuous operators ls : W 1,p(Qs) — W 1,p(Q) such that the sequence of norms ||ls|| is bounded and, for every s € N and for every v € W 1,p(Qs), we have qs(lsv) = v a.e. in Qs. Let, for every s € N, ws € W 1,p(Qs). Assume that the sequence of norms ||ws||Wi,p(Qs) is bounded. Then there exist an increasing sequence {sj} c N and a function w € W 1,p(Q) such that Is,wsj. — w weakly in W 1,p(Q), ^wSj — w a.e. in Q, and |wsj - ^w||LP(Qsj) — 0.

Proof. The properties of the operators ls along with the boundedness of the sequence of norms ||ws||wi,p(Qs) imply that the sequence {lsws} is bounded in W 1,p(Q) and

Vs € N, qs(lsws) = ws a.e. in Qs. (16)

Since the space W 1,p(Q) is reflexive and the sequence {lsws} is bounded in W 1,p(Q), there exist an increasing sequence {sk} c N and a function w € W 1,p(Q) such that w^fc — w weakly in W 1,p(Q). Hence, by condition (*1) of Theorem 1, we have w^, — w strongly in Lp(Q). Therefore, there exists an increasing sequence {sj} c {sk} such that lsjwsj — w a.e. in Q. It is clear that lsjwsj — w weakly in W 1,p(Q) and lsjwsj — w strongly in Lp(Q). The latter convergence along with (16) implies that ||wsj — qsjw||Lp(Qs,) — 0. □

Proposition 5. Let condition (*1) of Theorem 1 be satisfied, and assume that there exists a sequence of linear continuous operators ls : W 1,p(Qs) ^ W 1,p(Q) such that the sequence of norms ||ls|| is bounded and, for every s € N and for every v € W 1,p(Qs), we have qs(lsv) = v a.e. in Qs. In addition, assume that condition (*') of Theorem 2 is satisfied. Let, for every s € N, ws € W 1,p(Qs), and let w € W1,p(Q). Assume that the sequence of norms ||ws ||Wi,p(Qs) is bounded, and ||ws — qsw||Lp(Qs) ^ 0. Then lsws ^ w weakly in W 1,p(Q).

Proof. The properties of the operators ls imply that the sequence {lsws} is bounded in W 1>p(Q) and

Vs € N, qs(lsws) = ws a.e. in Qs. (17)

Assume that the sequence {lsws} does not converge weakly to w in W 1,p(Q). Then there exist a functional g € (W 1,p(Q))*, a number e > 0, and an increasing sequence {sk} C N such that

Vk € N, |<g,l-fcw-fc) —<g,w)| > e. (18)

Since the space W 1,p(Q) is reflexive and the sequence {lsws} is bounded in W 1,p(Q), there exist an increasing sequence {sj} C {sk} and a function wo € W 1,p(Q) such that

Is,wsj ^ wo weakly in W 1p(Q). (19)

Hence, by condition (*1) of Theorem 1, we have lsjwsj ^ w0 strongly in Lp(Q). Then, in view of (17), we have ||wsj — qsjw0||Lp(Qs ) ^ 0. This and the assumption that ||ws — qsw||Lp(Qs) ^ 0 imply that ||qs(w — wo)||lp(Qsj) ^ 0. Consequently, liininf ||qs(w — wo)||lp(qs) = 0. From this

equality, condition (*') of Theorem 2, and Proposition 3, we derive that w = w0 a.e. in Q. Then, in view of (19), we have lsjwsj ^ w weakly in W 1,p(Q). However, this contradicts (18). The obtained contradiction proves that lsws ^ w weakly in W 1,p(Q). □

The following definition essentially is a particular case of Definition 5 in [6].

Definition 6. Let, for every s € N, Us be a nonempty set in W 1,p(Qs), and let U be a nonempty set in W 1,p(Q). We say that the sequence {Us} H-converges to the set U if the following conditions are satisfied:

(a) for every function v € U, there exists a sequence ws € Us such that sup ||ws||Wi,p(Qs) <

seN s

and ||ws — qsv||LP(Qs) ^ 0;

(b) for every sequence vs € Us such that sup ||vs|Wi,p(Qs) < there exist an increasing

seN s

sequence {sj} C N and a function v € U such that ||vsj — qsjv||Lp(Qs ) ^ 0.

Proposition 6. Let condition (*') of Theorem 2 be satisfied. Then a sequence of nonempty sets Us C W 1,p(Qs) may H-converge to only one nonempty set U C W 1,p(Q).

Proof. Assume that a sequence of nonempty sets Us C W 1,p(Qs) H-converges to nonempty sets U C W 1,p(Q) and V C W 1,p(Q). Let w € U. Since the sequence {Us} H-converges to the set U, there exists a sequence ws € Us such that sup |ws^w 1,P

(Qs) < and ||ws — qsw|Lp(qs) ^ 0.

seN

Since the sequence {Us} H-converges to the set V, for the sequence {ws}, there exist an increasing sequence {sj} C N and a function v € V such that ||wsj — qsjv||Lp(Qs ) ^ 0. This convergence along with the convergence ||ws — qsw||Lp(Qs) ^ 0 implies that ||qsj(v — w)||Lp(Qs,) ^ 0. Then, taking into account condition (*') of Theorem 2 and Proposition 3, we find that w = v a.e. in Q. Therefore,

in view of the inclusion v € V, we have w € V. Consequently, U C V .In the same way, we prove that V C U. Thus, U = V. □

Remark 1. In the proof of Proposition 6, concerning the considered sets in W 1,P(Q), we implicitly assumed that functions equivalent to elements of these sets belong to the same sets.

Proposition 7. Assume that the embedding of W 1,P(Q) into LP(Q) is compact and the sequence of spaces W 1,P(Qs) is strongly connected with the space W 1,P(Q). Then the sequence {W 1,P(Qs)} H-converges to the set W 1,P(Q).

Proof. Let v € W 1,P(Q). For every s € N, we set ws = qsv. Obviously, for every s € N, we have ws € W 1,P(Qs). It is also easy to see that sup ||ws||Wi,p(ns) < and ||ws — qsv||LP(Qs) ^ 0.

seN s s

Next, taking a sequence vs € W 1,P(Qs) such that sup ||vs||Wi,P(ns) < in view of the assump-

seN s

tions of this proposition, we deduce from Proposition 4 that there exist an increasing sequence {sj} C N and a functon v € W 1,P(Q) such that ||vsj — qsjv||LP(qs ) ^ 0. Now, by Definition 6, we conclude that the sequence {W 1,P(Qs)} H-converges to the set W 1,P(Q). □

We note that condition (*') of Theorem 2 is essential for the conclusion of Proposition 6. This is justified by the following simple example.

Example 3. Assume that Q is a Lipschitz domain. Then the embedding of W 1,P(Q) into LP(Q) is compact. Let B be a closed ball in Rn such that B C Q, and assume that, for every s € N, Qs = Q \ B. In view of the known extension results for Sobolev spaces (see, for instance, [25, Theorem 7.25]), there exists a linear continuous operator l : W 1,P(Q \ B) ^ W 1,P(Q) such that, for every function v € W 1,P(Q \ B), we have Iv = v in Q \ B. Setting, for every s € N, ls = l, we find that the sequence {ls} has all the properties described in Definition 3. Therefore, the sequence of spaces W 1,P(Qs) is strongly connected with the space W 1,P(Q). Thus, Proposition 7 implies that the sequence {W 1,P(Qs)} H-converges to the set W 1,P(Q). Now, let y and r be the center and the radius of the ball B, respectively, and let B0 = {x € Rn : |x — y| ^ r/2}. We define

U = {v € W 1>P(Q) : v = 0 a.e. in Bo}.

It is easy to see that, for every function v € U, there exists a sequence ws € W 1,P(Qs) such that sup || ws || W1 ,P

(ns) < and ||ws-qsv||LP(ns) ^ 0. Next, we fix an arbitrary sequence vs € W 1,P(Qs)

seN

such that sup ||vs||Wi,P(ns) < Since the sequence {W 1,P(Qs)} H-converges to the set W 1iP(Q),

seNs

there exist an increasing sequence {sj} C N and a function v € W 1,P(Q) such that

||vsj - qsjv||LP(ns.) ^ 0. (20)

Let ^ be a function in C0^(Q) such that 0 ^ <p ^ 1 in Q, ^ = 1 in B0, and ^ = 0 in Q \ B. We have v^> € W 1,P(Q). Then, since ^ = 1 in B0, we have v — v^> € U. Moreover, taking into account that = 0 in Q\B, we derive from (20) that ||vsj — qsj(v — v^>)||LP(Qs ) ^ 0. Now, we conclude that the sequence {W 1,P(Qs)} H-converges to the set U. Obviously, U = W 1,P(Q). It remains to observe that Q \ (J Qs = B. Hence, measiQ \ (J Qs) > 0. Consequently, condition (*') of Theorem 2 is

s=1 ^ s=1 '

not satisfied.

We now proceed to a more delicate question on the H-convergence of sets defined by bilateral constraints.

Proposition 8. Assume that conditions (*1) and (*2) of Theorem 1 and condition (*') of Theorem 2 are satisfied. Let p, 0 : Q —> R, and let <p ^ tp a.e. in Q. Let, for every s € N, Us = {v € W 1,p(Qs) : p < v < 0 a.e. in Qs}, and let U = {v € W 1>p(Q) : p < v < 0 a.e. in Q}. Assume that the set U is nonempty. Then the sequence {Us} H-converges to the set U.

Proof. Let v € U. For every s € N, we set ws = qsv. Obviously, for every s € N, we have ws € Us. It is also easy to see that sup ||ws||Wi,p(ns) < and ||ws — qsv||Lp(Qs) — 0.

seN s s

Next, we fix an arbitrary sequence vs € Us such that sup ||vs||wi,p(ns) < Since con-

seN s

dition (*2) of Theorem 1 is satisfied, there exists a sequence of linear continuous operators ls : W 1,p(Qs) — W 1,p(Q) such that the sequence of norms ||ls|| is bounded and, for every s € N and for every v € W 1,p(Qs), we have qs(lsv) = v a.e. in Qs. Then, taking into account that condition (*1) of Theorem 1 is satisfied, we derive from Proposition 4 that there exist an increasing sequence {sj} C N and a function w € W 1,p(Q) such that lsjvsj — w a.e. in Q and ||vsj — qsjw||Lp(Qs ) — 0. Let us show that p ^ w ^ 0 a.e. in Q. Since, for every s € N, we have vs € Us, there exists a set E' C Q of measure zero such that, for every s € N and for every x € Qs \ E', we have p(x) ^ vs(x) ^ 0(x). In addition, by the properties of the operators ls, there exists a set E'' C Q of measure zero such that, for every s € N and for every x € Qs \ E'', we have (lsvs)(x) = vs(x). It is clear that

s € N, x € Qs \ (E' U E'') p(x) ^ (lsvs)(x) ^ 0(x). (21)

Since ls.vs. — w a.e. in Q, there exists a set E''' C Q of measure zero such that

Vx € Q \ E''', (Is, vs,) (x) — w(x). (22)

i

Next, for every k € N, we set

E(k) = Q \ |J Qs,. In view of condition (*') of Theorem 2, for every j=fc

i

k € N, we have meas

E(k) = 0. Therefore, setting E = (J E(k), we have meas E = 0. Now, let

fc=1

x € Q \ (E' U E'' U E''' U E). We fix an arbitrary e > 0. Since x € Q \ E''', by (22), we have (lsj vsj )(x) — w(x). Consequently, there exists k € N such that

j € N, j ^ k |(lsjvs,)(x) — w(x)| < e. (23)

Since x € Q \ E, there exists j € N, j ^ k, such that x € Qsj. Then we derive from (21) and (23) that p(x) — e ^ w(x) ^ 0(x) + e. Hence, in view of the arbitrariness of e, we obtain the inequality p(x) ^ w(x) ^ 0(x). Therefore, p ^ w ^ 0 a.e. in Q. Then w € U. Thus, we have established that, for every sequence vs € Us such that sup ||vs||Wi,p(ns) < there exist an increasing sequence

seN s

{sj} C N and a function w € U such that ||vsj — qsjw||Lp(qs,) — 0.

We now conclude that the sequence {Us} H-converges to the set U. □

We note that condition (*') of Theorem 2 is essential for the conclusion of Proposition 8. This is justified by the following example.

Example 4- Assume that the domain Q and the sequence of domains Qs are the same as in Example 3. Then conditions (*1) and (*2) of Theorem 1 are satisfied but condition (*') of Theorem 2 is not satisfied. Let p : Q —> R be the function such that, for every p(x) = 0.

Moreover, let 0 : Q —> R be the function such that

( ) i 0 if x € B, 0(x) = [ 1 if x € Q \ B.

Obviously, p ^ 0 in Q. Let, for every s € N, Us = {v € W 1,p(Qs) : p ^ v ^ 0 a.e. in Qs}, and let U = {v € W 1,p(Q) : p ^ v ^ 0 a.e. in Q}. Clearly, the set U is nonempty. Thus, all the conditions of Proposition 8 are satisfied except for condition (*') of Theorem 2. At the same time, the sequence {Us} does not H-converge to the set U. In fact, suppose that the sequence {Us} H-converges to the set U. Then, taking the sequence vs € W 1,p(Qs) such that, for every s € N, vs = 1 in Qs, we find that there exist an increasing sequence {sj} C N and a function v € U such

o

that ||vsj — qsjv||LP(Qs ) ^ 0. Hence, v = 1 a.e. in Q \ B. Therefore, v — 1 € W 1,p(Q). Moreover, since v € U, we have v = 0 a.e. in B. Thus, |Vv| = 0 a.e. in Q. Then, fixing a number r such

o

that 1 < r < min{p, n} and taking into account that v — 1 € W1,r(Q), we apply the corresponding Sobolev inequality for the function v — 1 and find that v = 1 a.e. in Q. However, this contradicts the fact that v = 0 a.e. in B. The obtained contradiction proves that the sequence {Us} does not H-converge to the set U.

Although, in the general case, condition (*') of Theorem 2 is essential for the H-convergence of sets defined by bilateral constraints, in the case of regular constraints, this condition does not play any role for the H-convergence of the corresponding sets. We demonstrate this by proving the following result.

Proposition 9. Assume that conditions (*1) and (*2) of Theorem 1 are satisfied. Let p, 0 € W 1,p(Q), and let p ^ 0 a.e. in Q. Let, for every s € N, Us = {v € W 1,p(Qs) : p ^ v ^ 0 a.e. in Qs}, and let U = {v € W 1,p(Q) : p ^ v ^ 0 a.e. in Q}. Then the sequence {Us} H-converges to the set U.

P r o o f. As in the proof of Proposition 8, we establish that, for every function v € U, there exists a sequence ws € Us such that sup ||wsy w (ns) < and ||ws — qsv|LP(ns) ^ 0.

seN

Next, we fix an arbitrary sequence vs € Us such that sup ||vs||Wi,p(ns) < In view of

seN s

condition (*2) of Theorem 1, there exists a sequence of linear continuous operators 1s : W 1,p(Qs) ^ W 1,p(Q) such that the sequence of norms ||1s|| is bounded and

Vs € N, qs(1svs) = vs a.e. in Qs. (24)

It is easy to see that the sequence {1svs} is bounded in W 1,p(Q). For every s € N, we set

zs = min{max{1svs, p}, 0}.

We have {zs} C U and the sequence {zs} is bounded in W 1,p(Q). Moreover, using (24) and the inclusions vs € Us, we establish that

Vs € N, qszs = vs a.e. in Qs. (25)

Using the reflexivity of the space W 1,p(Q), the boundedness of the sequence {zs} in W 1,p(Q), and condition (*1) of Theorem 1, we find that there exist an increasing sequence {sj} C N and a function v € W 1,p(Q) such that

zsj ^ v strongly in Lp(Q) (26)

and zsj ^ v a.e. in Q. The latter limit relation along with the inclusion {zsj} C U implies that v € U. Finally, we derive from (25) and (26) that ||vsj — qsjv||Lp(qs ) ^ 0. Thus, we have established that, for every sequence vs € Us such that sup ||vs||wi,p(ns)

< there exist an

seN s

increasing sequence {sj} C N and a function v € U such that ||vsj — qsjv||Lp(Qs ) ^ 0.

We now conclude that the sequence {Us} H-converges to the set U. □

Remark 2. Concerning some notions of convergence of sets lying in the same space, see, for instance, [26, 27]. Our notion of H-convergence of sets lying generally in variable spaces differs from the notions of convergence of sets in the sense of Kuratowski [26, Section 29] and in the sense of Mosco [27, Definition 1.1] even in the case of sets belonging to the same space.

We give one more result involving condition (*') of Theorem 2.

Proposition 10. Let conditions (*i), (*2), (*4), and (*5) of Theorem 1 be satisfied. In addition, let condition (*') of Theorem 2 be satisfied. Then there exist positive constants b1 and b2 such that, for every function v € W 1,P(Q), we have (F + G)(v) ^ MMIWi,p(Q) — b2.

Proof. By condition (*2) of Theorem 1, there exists a sequence of linear continuous operators 1s : W 1,P(Qs) ^ W 1iP(Q) such that the sequence of norms ||1s|| is bounded and, for every s € N and for every v € W 1,P(Qs), we have qs(1sv) = v a.e. in Qs. We set A = sup ||1s||. It is not

seN

difficult to find that A is a real number such that A ^ 1. Next, let v € W 1,P(Q). By virtue of condition (*4) of Theorem 1, there exists a sequence ws € W 1,P(Qs) such that ||ws — qsv||Lp(Qs) ^ 0 and Fs(ws) ^ F(v). The first of these limit relations and condition (*5) of Theorem 1 imply that Gs(ws) ^ G(v). Thus,

(Fs + Gs)(ws) ^ (F + G)(v). (27)

In view of (7), we have

Vs € N, (Fs + Gs)(ws) ^ C5||ws||Wi,p(Qs) — c6- (28)

This along with (27) implies that the sequence of norms ||ws||Wi,p(Qs) is bounded. Now, since condition (*1 ) of Theorem 1 and condition (*') of Theorem 2 are satisfied, we deduce from Proposition 5 that 1sws ^ v weakly in W 1,P(Q). Therefore,

liminf ||1sWs||wi,p(Q) ^ ||v|wi,p(Q). (29)

Moreover, we have

Vs € N, ||1sWs|wi,p(Q) ^ A||Ws||wi,p(Qs)• (30)

From (27)-(30), we derive that (F + G)(v) ^ C5A-P||v||Wi,p(Q) — □

We observe that condition (*') of Theorem 2 is essential for the conclusion of Proposition 10. In this regard, see [10, Example 4.3].

We complete the exposition of the results related to condition (*') of Theorem 2 with the following proposition.

Proposition 11. Assume that c > 0 and, for every open set H of Rn such that H c Q, we have liminf meas(Hn Qs) ^ cmeasH. Then condition (*') of Theorem 2 is satisfied.

s^-œ

Concerning the proof of this result, see, for instance, [10]. We also remark that the condition of Proposition 11 is satisfied in the case where the domains Qs have a perforated structure of the same kind as the structure of the domains considered in [16, Section 2].

Finally, we note that condition (*'') of Theorem 2 is also important for the conclusion of this theorem. In this regard, see [10, Example 4.4]. Obviously, condition (*'') of Theorem 2 is satisfied if all the functions are zero in the corresponding domains or if, for instance, for every s € N, we have = as^|Qs, where {as} c [0, +œ), as ^ 0, and ^ is a nonnegative function in L1(Q).

4. Conclusion

In this paper, we have formulated and have discussed some results on the convergence of sequences of minimizers and minimum values of functionals Fs + Gs : W 1,p(Qs) — R on sets of functions defined by bilateral constraints in domains Qs. These domains are assumed to be contained in a bounded domain Q of Rn. The functionals Fs are integral and convex, and their integrands satisfy the bilateral estimate c1|£|p — ^s(x) ^ /s(x,£) ^ c2|£|p + ^s(x) for almost every x € Qs and for every £ € Rn, where c1 and c2 are positive constants and are nonnegative functions such that the sequence of norms ||^s||Li(ns) is bounded. The functionals Gs are assumed to be weakly continuous on the corresponding Sobolev spaces. They are generally not integral and play a subordinate role.

We have considered two cases: the case of regular constraints, i.e., constraints lying in the Sobolev space W 1,p(Q), and the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function. In both cases, a certain connection of the spaces W 1,p(Qs) with the space W 1,p(Q), the T-convergence of the functionals Fs, and a convergence of the functionals Gs are essentially used. At the same time, each of these cases has a distinctive feature. In the first case, it is required that the difference between the upper and lower constraints be positive almost everywhere. In the second case, this requirement is absent. However, in the latter case, it is assumed that ||^s||Li(ns) — 0 and it is required that the exhaustion condition of the domain Q by the domains Qs be satisfied.

We have given a series of results involving the exhaustion condition. In particular, we have obtained an equivalent statement of this condition and, using it, have proved the H-convergence of sets of functions defined by bilateral (generally irregular) constraints in the domains Qs.

Acknowledgements

This work was supported by the Program of the Ural Branch of the Russian Academy of Sciences "Current Problems in Algebra, Analysis, and the Theory of Dynamic Systems with Applications to the Control of Complex Objects" (project "Development of New Analytic, Numerical, and Asymptotic Methods for Problems of Mathematical Physics and Applications to Signal Processing") and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

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