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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 376179, 15 pages
http://dx.doi.org/10.1155/2012/376179
Research Article

An -Estimate for Weak Solutions of Elliptic Equations

Dipartimento di Matematica, Università di Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Received 26 January 2012; Accepted 3 April 2012

Academic Editor: Svatoslav Staněk

Copyright © 2012 Sara Monsurrò and Maria Transirico. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove an -a priori bound, , for solutions of second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

1. Introduction

The Dirichlet problem for second order linear elliptic partial differential equations in divergence form and with discontinuous coefficients in bounded open subsets of , , is a classical problem that has been widely studied by several authors (we refer, e.g., to [16]).

In this paper, we want to analyze certain aspects of the same kind of problem, but in the framework of unbounded domains.

More precisely, given an unbounded open subset of , , we are interested in the study of the elliptic second order linear differential operator in variational form

with coefficients , and in the following associated Dirichlet problem Starting from a work of Bottaro and Marina (see [7]), who proved an existence and uniqueness theorem for the solution of problem (1.2), for , assuming that analogous results have been successively obtained weakening the hypotheses on the lower order terms coefficients. First generalizations in this direction have been carried on in [8], where and , , and satisfy assumptions similar to those in (1.3), but only locally. While in [9], for , these results have been further improved, since , , and are assumed to belong to opportune Morrey type functional spaces with lower summability.

In the above-mentioned works ([79]), the authors also provide the estimate

where the dependence of the constant on the data of the problem is completely described.

Here we suppose that the lower order terms coefficients are as in [9] for and as in [8] for and we prove an -a priori bound, . More precisely, for a sufficiently regular set and given a datum , we show that there exists a constant such that for any bounded solution of (1.2) and for every . We point out that also in our analysis the dependence of the constant is fully determined.

We note that bound (1.5) can be also useful when dealing with certain nonvariational problems that, by means of the existence of the derivatives of the , can be rewritten in variational form.

Among the authors who studied the Dirichlet problem for second order linear elliptic equations in divergence form with discontinuous coefficients in unbounded domains, we quote here also Lions in [10, 11] and Chicco and Venturino in [12].

The proof of (1.5) is developed as follows. In Section 2 we extend a known result by Stampacchia (see [1], or [13] for details), obtained within the framework of the generalization of the study of certain elliptic equations in divergence form with discontinuous coefficients on a bounded open subset of to some problems arising for harmonic or subharmonic functions in the theory of potential.

This is done in order to obtain a preliminary lemma, proved in Section 3, that permits to consider some particular test functions in the variational formulation of our problem. This allows us to prove a technical result (Lemma 4.1), that is the main point in the proof of the claimed -estimate.

2. A Generalization of a Result by Stampacchia

Let be a uniformly Lipschitz real function, such that there exists a positive constant such that for every one has and suppose that and that its derivative has a finite number of discontinuity points.

A known result by Stampacchia, see Lemma 1.1 in [1] (or in [13], for details), guarantees that given a function , defined in an open bounded subset of and belonging to , also the composition between and is in and gives an explicit expression for the derivative of this composition, up to sets of null Lebesgue measure.

Later on, in [7], Bottaro and Marina explicitly observed that, up to few modifications, the proof of these results remains valid also for an unbounded open subset of , . More precisely, and moreover In Lemma 2.2 below, we show a further generalization of (2.4), always in the case of unbounded domains.

In order to prove Lemma 2.2, we need the following convergence results.

Lemma 2.1. If has the uniform -regularity property, then for every , there exists a sequence of functions such that If is a uniformly Lipschitz function as in (2.1), (2.2), and (2.3) and such that its derivative has a finite number of discontinuity points, Moreover, there exists a sequence such that where with and .

Proof. The statement in (2.6) has been proved in [14].
The -convergence in (2.7) easily follows by (2.2) and by the convergence in (2.6). The -convergences in (2.8) and (2.9) can be obtained as in the proof of Lemma 1.1 of [13], with opportune modifications due to the fact that the set is unbounded (see also [7]).

We point out that next lemma is a fundamental tool in our analysis since it is the core of the proof of Lemma 3.3 that will allow us to take some specific test functions in the variational formulation of our problem.

This will consent to show a technical result (see Lemma 4.1), which is the main point in the proof of our -a priori bound.

Lemma 2.2. Let be a uniformly Lipschitz function as in (2.1), (2.2), and (2.3) and such that its derivative has a finite number of discontinuity points. If has the uniform -regularity property, then for every one has

Proof. Fix ; to show (2.10) we need different arguments according to different values of .
For we need to verify that there exists a positive constant such that this ends the proof of our lemma as a consequence of a characterization of the space (see, e.g., Proposition IX.18 of [15]).
In order to prove (2.11), we consider the sequence introduced in Lemma 2.1 and observe that, given , one has .
Indeed, by Hölder inequality we get and this quantity vanishes letting , as a consequence of (2.6) and (2.7).
On the other hand, Having in mind (2.12), we want to pass to the limit as in the right-hand side of this equality.
Concerning the first term, by (2.2), Hölder inequality, and using the last relation in (2.6), we obtain Thus, by the convergence in (2.6), the quantity on the left-hand side goes to zero, letting , and therefore For the last term we have Indeed, Moreover, by the weak convergence in (2.8) the first term on the right-hand side vanishes letting . Concerning the second one, we get and, by (2.6) and (2.8), also this quantity is null passing to the limit as .
It remains to treat the second term of the right-hand side of (2.14). To this aim let us introduce the sets where is that of (2.3).
We observe that, in view of (2.6), there exists such that, up to sets of null Lebesgue measure, and we can assume, without loss of generality, that .
Therefore, by (2.3) and (2.21), one has On the other hand, always using (2.6), we can also deduce, with no loss of generality, that This, together with (2.6) and (2.7), and by definition of , gives, up to a subsequence, Moreover, by (2.2) and (2.6), .
Therefore, (2.24) and (2.25) being true, the bounded convergence theorem applies giving, up to a subsequence, Combining (2.12), (2.14), (2.16), (2.17), and (2.26), we conclude, by (2.2) and Hölder inequality, that for , that is (2.11).
For , let us consider the sequence introduced in Lemma 2.1 and put Simple calculations give with positive constant depending only on .
We want to pass to the limit in the right-hand side of this inequality. For the first term it is easily seen that it goes to zero, in view of (2.9).
For the last one, again from (2.9), we get, up to a subsequence, Moreover, by (2.2) it follows that Hence, from these last considerations and using the bounded convergence theorem we obtain, up to a subsequence, Therefore, by (2.29), up to a subsequence, we have Now, observe that , because of its compact support, then for any there exists a sequence such that this means that there exists such that By (2.33) and (2.35) we deduce that this ends the proof of our lemma.

3. Tools

We recall the definitions of the Morrey type spaces where the lower order terms coefficients of the operator will be chosen. These functional spaces were introduced for the first time in [9] in order to generalize to the case of unbounded domains of the classical notion of Morrey spaces.

We start with some notation. Given any Lebesgue measurable subset of , we denote by the -algebra of all Lebesgue measurable subsets of . For any , is its characteristic function and is the intersection , where is the open ball centered in and with radius .

For and , the space of Morrey type is the set of all the functions in such that

endowed with the norm just defined. Moreover, denotes the closure of in .

For reader’s convenience, we state here a result of [16], adapted to our needs, providing the boundedness and an embedding estimate for the multiplication operator where the function belongs to a suitable space of Morrey type .

Lemma 3.1. If , with , if , and , if , then the operator in (3.2) is bounded. Moreover, there exists a constant such that with .

Now, we recall a lemma, proved in [9], describing the main properties of some functions , introduced in [7], that will be of crucial relevance in the proof of our main result.

Let and , with . For each we set

Lemma 3.2. Let , and . Then there exist and , with , such that, setting one has and with positive constant.

As already mentioned, the next lemma will allow us, in the last section, to take the products as test functions in the variational formulation of our problem.

Lemma 3.3. If has the uniform -regularity property, then for every and for any one has where , for , are the functions of Lemma 3.2.

Proof. If , then ; therefore, by Lemma 3.2 in [14], one has .
If and , then , therefore , for the choice in (2.3). This entails that , by means of Lemma 2.2.
In view of these considerations and (3.11) being true, we also get .

4. An A Priori Bound

Let be an unbounded open subset of , , such that We consider in the second order linear differential operator in variational form with the following conditions on the coefficients: We also associate to the bilinear form.

We point out that, as a consequence of Lemma 3.1, is continuous on and so the operator is continuous as well.

We start showing a technical lemma.

Let be the functions of Lemma 3.2 obtained in correspondence of a given , of and of a positive real number specified in the proof of Lemma 4.1. The following result holds true.

Lemma 4.1. Let be the bilinear form defined in (4.2). Under hypotheses (21)–(24), there exists a constant such that with .

Proof. Let , , , and , for , be as above specified and . We start observing that in view of Lemma 3.3 one has , for .
This allows us to take as test function in (4.2). Hence, simple calculations together with (3.9) and (3.10) give From this last equality, (3.8), and hypotheses (21) and (24) we get Hence, setting we obtain On the other hand, by the Hölder inequality, Lemmas 3.2 and 3.3, the embedding results contained in Lemma 3.1 and using hypothesis (23) and (3.7), one has that there exists a constant , such that with .
Now, we observe that explicit computations give with .
Therefore, combining (4.8), (4.9), and (4.10) we get with .
Thus, with .
Choosing and , we have Finally we conclude by (4.7) and (4.13) that with . This ends the proof of (4.3).

Finally, we consider the Dirichlet problem and we prove the following -a priori bound.

Theorem 4.2. Under the hypotheses (21)–(24) and if is in and the solution of (4.15) is in , then one has where is a constant depending on , .

Proof. Fix . We firstly prove that with .
Indeed, if we consider the functions , , obtained in correspondence with the solution , of and as in Lemma 4.1, by (3.11) we get with .
Thus, taking into account (4.3), with and .
The linearity of together with (3.11) and (3.12) then give (4.17).
Now, using (4.17) and Hölder inequality we end the proof, since

References

  1. G. Stampacchia, “Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,” Annales de l'Institut Fourier, vol. 15, no. 1, pp. 189–258, 1965. View at Google Scholar
  2. O. A. Ladyzhenskaja and N. N. Ural'tzeva, Equations aux Derivèes Partielles de Type Elliptique, Dunod, Paris, France, 1966.
  3. C. Miranda, “Alcune osservazioni sulla maggiorazione in Lv delle soluzioni deboli delle equazioni ellittiche del secondo ordine,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 61, pp. 151–169, 1963. View at Google Scholar
  4. M. Chicco, “An a priori inequality concerning elliptic second order partial differential equations of variational type,” Le Matematiche, vol. 26, pp. 173–182, 1971. View at Google Scholar
  5. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, Springer, Berlin, Germany, 2nd edition, 1983.
  6. N. S. Trudinger, “Linear elliptic operators with measurable coefficients,” Annali della Scuola Normale Superiore di Pisa 3, vol. 27, pp. 265–308, 1973. View at Google Scholar
  7. G. Bottaro and M. E. Marina, “Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati,” Bollettino Unione Matematica Italiana Serie 4, vol. 8, pp. 46–56, 1973. View at Google Scholar
  8. M. Transirico and M. Troisi, “Equazioni ellittiche del secondo ordine a coefficienti discontinui e di tipo variazionale in aperti non limitati,” Bollettino Unione Matematica Italiana Serie 7, vol. 2, pp. 385–398, 1988. View at Google Scholar
  9. M. Transirico, M. Troisi, and A. Vitolo, “Spaces of Morrey type and elliptic equations in divergence form on unbounded domains,” Bollettino Unione Matematica Italiana Serie 7, vol. 9, no. 1, pp. 153–174, 1995. View at Google Scholar
  10. P. L. Lions, “Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés,” Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 8, vol. 78, no. 5, pp. 205–212, 1985. View at Google Scholar
  11. P. L. Lions, “Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés. II,” Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 8, vol. 79, no. 6, pp. 178–183, 1985. View at Google Scholar
  12. M. Chicco and M. Venturino, “Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 178, pp. 325–338, 2000. View at Publisher · View at Google Scholar
  13. G. Stampacchia, Equations Elliptiques du Second Ordre à Coefficients Discontinus, Séminaire de Mathématiques Supérieures, 4e session, été 1965, Les Presses de l'Université de Montréal, Montreal, Que, 1966.
  14. L. Caso, P. Cavaliere, and M. Transirico, “Solvability of the Dirichlet problem in W2,p for elliptic equations with discontinuous coefficients in unbounded domains,” Le Matematiche, vol. 57, no. 2, pp. 287–302, 2002. View at Google Scholar
  15. H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, France, 1983.
  16. P. Cavaliere, M. Longobardi, and A. Vitolo, “Imbedding estimates and elliptic equations with discontinuous coefficients in unbounded domains,” Le Matematiche, vol. 51, no. 1, pp. 87–104, 1996. View at Google Scholar