Weak convergence under nonlinearities

Abstracts

In this paper, we prove that if a Nemytskii operator maps Lp(omega, E) into Lq(omega, F), for p, q greater than 1, E, F separable Banach spaces and F reflexive, then a sequence that converge weakly and a.e. is sent to a weakly convergent sequence. We give a counterexample proving that if q = 1 and p is greater than 1 we may not have weak sequential continuity of such operator. However, we prove that if p = q = 1, then a weakly convergent sequence that converges a.e. is mapped into a weakly convergent sequence by a Nemytskii operator. We show an application of the weak continuity of the Nemytskii operators by solving a nonlinear functional equation on W1,p(omega), providing the weak continuity of some kind of resolvent operator associated to it and getting a regularity result for such solution.

weak continuity; nonlinearities; Nemytskii operator

Neste artigo, provamos que se um operador de Nemytskii aplica Lp(ômega, E) no Lq(ômega , F), para p, q maiores do que 1, E, F espaços de Banach separáveis e F reflexivo, então uma seqüência que converge fracamente e q.t.p. é transformada em uma seqüência fracamente convergente. Fornecemos um contra-exemplo mostrando que se q = 1 e p é maior do que 1, podemos não ter continuidade seqüêncial de tal operador. Contudo provamos que se p = q = 1, então seqüências fracamente convergentes que convergem q.t.p. são aplicadas em seqüências fracamente convergentes por um operador de Nemytskii. Mostramos uma aplicação da continuidade fraca dos operadores de Nemytskii resolvendo uma equação funcional não linear no W1,p(ômega), provando a continuidade fraca de um tipo de operador resolvente associado ao operador de Nemytskii e obtendo um resultado de regularidade de tal solução.

continuidade fraca; não linearidades; operador de Nemytskii

Weak convergence under nonlinearities

DIEGO R. MOREIRA^I; EDUARDO V. O. TEIXEIRA^II

^IDepartamento de Matemática, Universidade Federal do Ceará, Campus do Pici

60455-760 CE, Brazil

^IIDepartment of Mathematics, University of Texas at Austin, RLM 9.136

Austin, Texas, 78712-1082 U.S.A.

^{Correspondence}

ABSTRACT

In this paper, we prove that if a Nemytskii operator maps Lp(W, E) into Lq(W, F), for p, q greater than 1, E, F separable Banach spaces and F reflexive, then a sequence that converge weakly and a.e. is sent to a weakly convergent sequence. We give a counterexample proving that if q = 1 and p is greater than 1 we may not have weak sequential continuity of such operator. However, we prove that if p = q = 1, then a weakly convergent sequence that converges a.e. is mapped into a weakly convergent sequence by a Nemytskii operator. We show an application of the weak continuity of the Nemytskii operators by solving a nonlinear functional equation on W1,p(W), providing the weak continuity of some kind of resolvent operator associated to it and getting a regularity result for such solution.

Key words: weak continuity, nonlinearities, Nemytskii operator.

RESUMO

Neste artigo, provamos que se um operador de Nemytskii aplica Lp(W, E) no Lq(W , F), para p, q maiores do que 1, E, F espaços de Banach separáveis e F reflexivo, então uma seqüência que converge fracamente e q.t.p. é transformada em uma seqüência fracamente convergente. Fornecemos um contra-exemplo mostrando que se q = 1 e p é maior do que 1, podemos não ter continuidade seqüêncial de tal operador. Contudo provamos que se p = q = 1, então seqüências fracamente convergentes que convergem q.t.p. são aplicadas em seqüências fracamente convergentes por um operador de Nemytskii. Mostramos uma aplicação da continuidade fraca dos operadores de Nemytskii resolvendo uma equação funcional não linear no W1,p(W), provando a continuidade fraca de um tipo de operador resolvente associado ao operador de Nemytskii e obtendo um resultado de regularidade de tal solução.

Palavras-chave: continuidade fraca, não linearidades, operador de Nemytskii.

1. INTRODUCTION

A very important question in Functional Analysis is how to decide if an operator in a Banach space is weakly continuous. Frequently, we meet this issue in variational problems when we have to check the main assumptions of the classical theorems, especially if we are searching for some kind of compactness results. This question becomes more difficult when we deal with nonlinearities. Among the nonlinear operators, there is an outstanding group called Nemytskii operators. We are interested in the weak sequential continuity of these operators. In order to develop these ideas, we consider the notion of a.e. and weak convergence (a.e.w.) and formulate the problem a_p,q where p, q ³ 1 to be: Let f be a Caratheodory function and suppose that the Nemytskii operator associated to f maps L^p(W,) intoL^q(W,). Does N_f map a.e.w. convergent sequences into a.e.w. convergent sequences? Our goal in this paper is to study under what conditions the problem a_p,q is affirmatively answered.

Our paper is organized as follows. In the section 2, we treat the heart of the matter. We start providing a uniqueness result of convergence in the a.e.w sense for L^p(W,) spaces (Lemma 1). Afterwards, we construct an example which shows the affirmative answer for the problem a_{p, 1} fails to p > 1 (Example 2) and we establish the solvability of the problem a_p,q under the following assumptions: q > 1 and reflexivity of (Theorem 2). At the end of this section, we prove the counterpart of the example 2 which says that the problem a_{1, 1} is solvable on bounded domains (Theorem 4). In the section 3, the last section, we are concerned about studying the solvability on W^{1, p}(W) of the equation

f (x, u(x)) - lu(x) = y(x)

for l Î and y Î W^{1, p}(W) given. We also provide conditions to the weak sequential continuity of the resolvent operator R_l = (N_f - l I )^-1 on W^{1, p}(W) and we observe a regularity result for such solutions. In the study of the problems a_p,q, surprisingly, the cases q = 1 and q > 1 have been shown very different. Some of these facts turned out to be known, mainly in particular cases; however not in such a generality. We think it is worthwhile to formulate them in a more general form and make them more available. We believe the ideas developed in this paper may be applied in quite different problems.

2. WEAK CONTINUITY OF THE NEMYTSKII OPERATOR

DEFINITION 1. Let W be a domain in ^N. Let and be separable Banach spaces. A function f : W x ® is said to be a Caratheodory function if:

(a) for each fixed v Î the function x f (x, v) is Lebesgue measurable in W;

(b) for almost everywhere fixed x Î W the function f (x,^.) : ® is continuous.

In this case we denote f Î (C). Let (W,) be all measurable functions u : W ® . It is easy to prove that if f Î (C) then f defines a mapping N_f : (W,) ® (W,) by N_f(u)(x) : = f (x, u(x)). This mapping is called the Nemytskii operator associated to f. The first result we would like to state is an extention to separable Banach spaces of the remarkable theorem due to Vainberg concerning about the Nemytskii operator theory.

THEOREM 1 (Lucchetti and Patrone 1980). Let , be separable Banach spaces. The Nemytskii operator N_f maps L^p(W,) into L^q(W,), 1 £ p, q < + , if and only if there exist a constant a > 0 and b(x) Î L^q₊(W) such that

In this case, the operator N_f is continuous and bounded, in the sense that maps bounded sets in bounded sets

DEFINITION 2. Let (X,,m) be a measure space, (X,) a topological vector space of -valued functions defined on X and (f_n)_{n ³ 1}Ì (X,) . We said f_n ® (f, g) a.e.w.(almost everywhere and weakly) in (X,) if f_n ® f a.e. in X and f_n

g in (X,). If f = g we just say f_n ® f a.e.w. in (X,).

The next Lemma gives a kind of uniqueness of the limit in the above convergence in L^p(X,m,) spaces.

LEMMA 1 (Moreira 2001, Teixeira 2001). Let (X,,m) be a -finite measure space and (u_n)_{n ³ 1}Ì L^p(X,m,), 1 £ p < + . Suppose that u_n ® (u, v) a.e.w. in L^p(X,m,). Then u = v, and therefore u_n ® u a.e.w. in L^p(X,m,).

PROOF. There exists a sequence {X_j} of measurable subsets of X such that:

Let j ³ 1 be fixed. Given e₁ = 1, by Egorov's theorem, there is a subset A₁ of X_j,m(A₁) < 1 such that u_n® u in L(X_j \ A₁,), in particular,

Thus we have u = v a.e. in X_j \ A₁. Taking now e₂ = and applying Egorov's theorem again, we obtain a subset A₂ of A₁, m(A₂) < such that u_n ® u in L(A₁ \ A₂,) hence u_n ® u in L^p(X_j \ A₂,) and therefore, we have u = v a.e. in X_j \ A₂. Carry on this process we get a decreasing sequence {A_n}, m(A_n) < and u = v a.e. in X_j \ A_n. Set AX_j = A_n. This way m(AX_j) = 0 and u = v a.e. in X_j \ AX_j. To finish, we define

thus

This concludes the Lemma.

EXAMPLE 1 (Teixeira 2001).Let 1 < p < + , m Î and let W be a domain in ^N. Every bounded sequence in W^{m, p}(W) contains a subsequence that converges a.e.w. to some function in W^{m, p}(W). Indeed, suppose (u_n)_{n ³ 1}Ì W^{m, p}(W), || u_n||_{W^{m, p}(W)}£ C. Since W^{m, p}(W) is reflexive, we can suppose that u_n

u Î W^{m, p}(W). Let be a nested sequence of relatively compact open sets exhausting W. For all j ³ 1 let v_j Î C(^N;) be a function holding the following properties: 0 ³ v_j ³ 1, v_j 1 in , v_j 0 in ^N \ . By Sard's theorem, there exists a regular value 0 < a < 1 of v_j. Let = (a, + ). So {} is a nested sequence of C relatively compact open sets of W. Since the embedding W^{m, p}() L^p() is compact, u_n® u in L^p(); therefore, for each j ³ 1, (u_n)_{n ³ 1} has a subsequence converging a.e. to u in . Using the Cantor Diagonal Argument, we build a subsequence of (u_n)_{n ³ 1} that converges a.e. in W to u.

We are interested in the following problem: When does the Nemytskii operator map a.e.w. convergent sequences into a.e.w. convergent sequences? This question is a way of asking about the weak sequential continuity of the Nemytskii operator. More precisely, our problem is

Of course, a.e. convergent sequences are mapped into a.e. convergent sequences by a Nemytskii operator. Actually, what we want to know is when this class of operator maps a.e.w. convergent sequences into weakly convergent sequences.

It is reasonable to suspect that the problem a_{p, 1}, 1 < p < + , cannot be affirmatively answered because if it were solvable, we would automatically get, without domain dependence, that the embedding W^{1, p}(W) L^p(W) would be compact. However, there exist many domains where we have lack of compactness of such an embedding. The next example shows this directly.

EXAMPLE 2.Let 1 < p < + and W be a domain in ^N. Then answer of the problem a_{p, 1} is negative. Indeed, without lost generality we may assume 0 Î W. Set f : W x ® , f (x, t) = |t|^{p .} v₀, where v₀Î \ {0}. The Nemytskii operator N_f applies L^p(W) into L¹(W,). Set (u_n)_n³1Ì L^p(W), u_n = |B_n|^-1/p, where B_n = {x Î ^N;|x| < 1/n}. Since u_n ® 0 a.e. in W, ||u_n||_p = 1 and L^p(W) is reflexive, we may assume u_n ® 0 a.e.w. It is easy to check that N_f(u_n) 0 in L¹ (W,). In fact, from Hahn-Banach theorem, there exists a y Î ^* such that y(v₀) = 1. Define Î [L¹(W,)]^*, setting

we obtain

But if we have the presence of the reflexivity, the situation changes and we get the following very useful result. The next theorem is an improvement of the result found in (Moreira 2001, Teixeira 2001).

THEOREM 2 (Moreira 2001, Teixeira 2001). Let 1 ³ p, q < + with q 1 and W be a domain in ^N. If is reflexive, then answer of the problem a_p,q is affirmative.

PROOF. By theorem 1, the Nemytskii operator N_f : L^p(W,) ® L^q(W,) is a bounded map. Suppose u_m® u a.e.w. in L^p(W,). Since (u_m) is bounded in L^p(W,), (N_f(u_m)) is bounded in L^q(W,). By reflexivity, we can extract a subsequence N_f(um_k) v Î L^q(W,). Clearly, N_f(um_k) ® N_f(u) a.e. in W. Therefore, by Lemma 1, N_f(um_k) ® N_f(u) a.e.w. in L^q(W,). So far, we have proven that if u_m® u a.e.w. in L^p(W,) there exists a subsequence (um_k) of (u_m) such that N_f(um_k) ® N_f(u) a.e.w. in L^q(W,). We claim that

In fact, as we have already observed, we only need to show that N_f(u_m) N_f(u) in L^q(W,). Suppose, by a contradiction, this is not the case. Thus there is a weak neighborhood N^w(N_f(u)) of N_f(u) and a subsequence (um_j), N_f(um_j) N^w(N_f(u)) j ³ 1. Naturally, um_j® u a.e.w., then applying the first step of this proof, we obtain a subsubsequence (umj_k) of (um_j), N_f(umj_k) ® N_f(u) a.e.w. in L^q(W,), a contradiction, since N_f(umj_k) N_f(u) because N_f(umj_k) N^w(N_f(u)) k ³ 1.

COROLLARY 1. If W is a domain in ^N and u_m ® u a.e.w. in L^p(W), 1 < p < + . Then u_m⁺ ® u⁺, u_m^- ® u^-, | u_m| ® | u| all these convergences being in the a.e.w. sense in L^p(W).

COROLLARY 2. If W is a domain in ^N, m Î and N_f maps the L^p(W) into L^q(W), 1 < p, q < + , the operator N_f : W^{m, p}(W) ® L^q(W) is weakly sequentially continuous.

COROLLARY 3. Let W be a bounded domain in ^N. If u_n ® u a.e.w in L^p(W) with p > 1, then for all 1 £ q < p, u_n ® u in L^q(W). Consequently, W^{1, p}(W) is compactly embedded in L^q(W), for all 1 £ q < p, without any regularity condition on W.

PROOF. Let us fix 0 < < p - 1. Since L^p(W) L^{p - e}(W) we have that u_n

u in L^{p - e}(W). Set f : ® given by f (t) = | t|^{p - e}. N_f maps L^p(W) into L^{p/p - e}(W). From theorem 2, f (u_n) f (u) in L^{p/p - e}(W). In particular

So we have that u_n

u in L^{p - e}(W) and ||u_n||_{L^{p - e}(W)}® ||u||_{L^{p - e}(W)}. Once L^{p - e} is uniformly convex, it implies that u_n® u in L^{p - e}(W). Moreover, using again the fact that L^r(W) is continuosly embedded into L^s(W) for r ³ s, we are able to conclude that u_n® u in L^q(W) for all q Î [1, p - ]. Letting ® 0, we get the result. The second part of the corollary follows from the first part added with example 1.

It is worthwhile to stand out that the corollary 3 is sharp. In general, W^{1, p}(W) L^q(W) for q ³ p. This fact can be found in (Adams 1975).

We remark that weak convergence in L^p(W) is not suffice to conclude the thesis of theorem 2. In fact, let W = (0,), u_n(x) = sin(nx) 0 in L²(0,), and let f : (0,) x ® be given by: f (x, s) = s⁺. Now we note that: 1, f (x, u_n(x)) = (sin(y))⁺dy, so:

It remains to study the problem a_{1, 1}. In order to start analyzing this problem, we shall state the general version of Dunford-Pettis theorem, obtained by Talagrand in 1984.

THEOREM 3 (Talagrand 1984). Let W be a bounded domain in ^N and be a weak complete Banach space. Let Ì L¹(W,) be a bounded convex subset. Then is weakly relatively compact, if and only if it satisfies the following two conditions:

1. {||

   ||

   

   W ®

   
   

   Î

   } Ì

   L

   ¹(W)

   is weakly precompact;

2. for each sequence (

   )

   in

   ,

   the set of

   x Î W

   such that there is a

   k for which the sequence (

   )

   _n³k is equivalent to the vector basis of l¹ has measure zero.

Let us point out that from Dunford-Pettis's theorem, the condition 1 above is equivalent to the equiintegrability of {||||

Î }. The next theorem is the counterpart of example 2.

THEOREM 4. Let W be a bounded domain in ^N. If is reflexive, then the answer of the problem a_{1, 1} is affirmative.

PROOF. Let u_m® u a.e.w. in L¹(W,). Defining u₀ = u, by the Eberlein-Smulian theorem (Brito 1998) the set K = {u_m;m ³ 0} is weakly compact, since it is weakly sequentially compact. Let us denote X = (K). From Krein's theorem (Brito 1998) we get that X is weakly compact, thus, in particular theorem 3 says that X is equiintegrable. The equiintegrability means that if > 0 is given; there exists d₁ > 0 such that

By theorem 1, the Caratheodory function f satisfies the following growth condition:

where a > 0 and b(x) Î L(W). Let Y = (N_f(K)). If v Î co(N_f(K)), there exist functions u₁,..., u_nÎ K and positive numbers ,..., fufilling l_j = 1 such that v = l_j ^. u_j. In this way

Since

b(x)dx = 0 there exists l₂ > 0, such that | A| < l₂

b(x)dx < ; therefore

Thus, we obtain

We have just verified the condition 1 of theorem 3, for Y. However, by hypothesis, the condition 2 we get for free, since being reflexive, it does not contain a copy of l¹; therefore by theorem 3, the set Y is weakly compact, and thus so is N_f(K) = {N_f(u_m) : m ³ 0}. Using again the Eberlein-Smulian theorem, we can extract a subsequence um_k, such that, N_f(um_k) v in L¹(W,). Since N_f(um_k) ® N_f(u) a.e. in W, by the Lemma 1, v = N_f(u) and then N_f(um_k) N_f(u) in L¹(W,). So far, we have proven that if u_m® u a.e.w in L¹(W,) there exists a subsequence (um_k) of (u_m) such that N_f(um_k) N_f(u) in L¹(W,). We can repeat the same argument used in the proof of theorem 2 and obtain N_f(u_m) N_f(u) in L¹(W,).

It is interesting to notice that follows immediately from theorem 4 if u_n® u a.e.w. in L¹(W,) then u_n® u in the L¹(W,)-norm topology.

3. AN APPLICATION

We shall provide an application of theorem 2 by solving a general nonlinear equation on the Sobolev spaces W^{1, p}(W). The problem studied here is a very natural question for the Nemytskii operator on Sobolev spaces. Indeed, the problem we shall work on is:

Let 1 < p < , let W be a bounded domain in ^N and let f : W x ® be a Lipschitzian function (In this paper, the Lipschtz norm is defined using the sum norm in euclidean space, i.e, |(x, s)|_{^N x} = |x|_^N + |s|), such that

for some a > 0 and some b Î L^p₊(W). Given a y Î W^{1, p}(W), and given a l Î we are interested in finding u Î W^{1, p}(W) such that

(P)

DEFINITION 3. Let f:W x ® be a Caratheodory function such that the Nemytskii operator N_f maps L^p(W,) into L^q(W,). We define (f)= inf{a > 0

b Î L

such that }.

Let us remark that the infimum on this definition actually is a minimum. Indeed, let a_n be a minimizing sequence for (f ), and let b_nÎ L^p(W) be functions such that the following inequality || f (x, s)||£ a_n|| s|| + b_n(x) holds for all n Î , s Î and a.e. x Î W. Taking the lim inf_n, we find || f (x, s)||£ (f )|| s|| + (x) where (x) = lim inf_nb_n(x).

THEOREM 5. The problem above is answered affirmatively for all l > ||f||_Lip. Moreover the solution is unique and the operator (N_f - lId)^-1 : W^{1, p}(W) ® W^{1, p}(W) is sequentially weakly continuous.

PROOF. Initially, we remark that (f) ³ ||f||_Lip. In fact, since

we get

Let us start by estimating || f (x,(x))||_{W^{1, p}(W)}:

This estimative above tell us N_f : W^{1, p}(W) ® W^{1, p}(W) is a bounded operator. Therefore, from the same argument found on the final step of theorem 2, we conclude N_f is sequentially weak continuous.

Let us define : W^{1, p}(W) ® W^{1, p}(W) by

We observe that once N_f is sequentially weak continuous, so is . Moreover, to solve (P) is equivalent to find a fixed point of .

Let us fix M > . For such a M we see that if ||||_{W^{1, p}(W)} < M

In other words, maps the ball of radius M in W^{1, p}(W) into itself, i.e., : B_{W^{1, p}}[M] ® B_{W^{1, p}}[M]. Let X denote B_{W^{1, p}}[M] endowed with the weak topology. So X is a compact convex subset of a locally convex space. In additional, as we pointed out before, : X ® X is a continuous map.

Finally, we can use the Leray-Schauder-Tychonoff fixed point theorem (Dunford and Schwartz 1964), and conclude that has a fixed point which is precisely a solution of (P). Now, let us suppose that there exist u₁, u₂Î W^{1, p}(W) such that

Subtracting these equations we find f (x, u₁(x)) - f (x, u₂(x)) = l(u₁(x) - u₂(x)). Therefore

If u₁(x) - u₂(x) 0, we would be able to cancel this expression at the inequality above and we would find, l ³ ||f||_Lip. Hence the solution of (P) is unique.

In order to study the weak continuity of = (N_f - lId )^-1 : W^{1, p}(W) ® W^{1, p}(W), we shall use the same idea found in the final step of theorem 2. Since all weakly convergent sequence in W^{1, p}(W) has a subsequence converging a.e., to prove is sequentially weakly continuous, it is enough to show that is bounded. Suppose (y) = u,

Then

Writing in a better way,

This estimative shows the operator is bounded.

The main information given by theorem 5 is the regularization of the solution. We observe that if we see the map , defined on the proof of this theorem, as : L^p(W) ® L^p(W), it is easy to verify that it is a contraction; therefore from the Banach Fixed Point theorem, the problem (P) has, for all y Î L^p(W), always a unique solution u Î L^p(W), provided l > ||f||_Lip. The main point of theorem 5 is that u Î W^{1, p}(W) whenever y Î W^{1, p}(W).

Let us point out that in a special case when f(x, s) = A|s| + b(x), b Î Lip(W), we can improve theorem 5, saying that (P) is solvable for all l > (f ). However we cannot expect to solve (P) if l ³ (f) as the following simple situation show us: Let f : W x ® be defined by f(x, s) = |s|. In this case, (f)= ||f||_Lip = 1. Suppose l ³ 1, then f(x, u(x)) - lu(x) = |u(x)| - lu(x) ³ 0. Hence if y Î W^{1, p}(W), with y(x) < 0, it is impossible to solve the equation (P).

From these comments, a interesting question arises in this problem: What can we say when (f) < l ³ ||f||_Lip?

4. ACKNOWLEDGMENTS

The second author would like to thank professor Haskell Rosenthal (University of Texas - Austin) for bringing the Talagrand's paper to his attention. The authors are grateful for the financial support by CAPES and CNPq, respectively, which are brazilian government agencies that finance the development of science and technology.

Manuscript received on May 31, 2002;

accepted for publication on October 25, 2002;

presented by JOÃO LUCAS BARBOSA

• ADAMS RA. 1975. Sobolev Spaces. Academic Press, New York, p. 122-161.
• BRITO W. 1998. Compacidad debil en espacios de Banach y aplicaciones de un teorema de R.C. James. Universidad de los Andes, Faculdad de Ciencias, Notas de Matematica, Serie Pre-Print, p. 2-42.
• DUNFORD N AND SCHWARTZ J. 1964. Linear operator, 2nd ed., Interscience Publishers, New York, p. 456.
• LUCCHETTI R AND PATRONE, F. 1980. On Nemytskii's operator and its application to the lower semicontinuity of integral functionals, Indiana Univ Math J 29(5), p. 703-735.
• MOREIRA D. 2001. Soluçőes positivas de equaçőes elípticas năo lineares na presença de expoentes críticos, Universidade Federal do Ceará, Master degree manuscript, p. 12-15.
• TALAGRAND M. 1984. Weak Cauchy sequence in L¹(E). Amer J Math 106: 703-724.
• TEIXEIRA E. 2001. O princípio de concentraçăo e compacidade e aplicaçőes, Universidade Federal do Ceará, Master degree manuscript, p. 36-43.

Publication Dates

• Publication in this collection
  17 Apr 2003
• Date of issue
  Mar 2003

History

• Accepted
  25 Oct 2002
• Received
  31 May 2002