International Journal of Differential EquationsVolume 2011 (2011), Article ID 572824, 17 pagesdoi:10.1155/2011/572824

Research Article

A Priori Bounds for a Class of Elliptic Operators

Sara Monsurrò, Maria Salvato, and Maria Transirico

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

Received 17 June 2011; Accepted 5 August 2011

Academic Editor: Charles E. Chidume

Copyright © 2011 Sara Monsurrò et al. 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 obtain some a priori bounds for a class of uniformly elliptic second-order differential operators, both in a no-weighted and in a weighted case. We deduce a uniqueness and existence theorem for the related Dirichlet problem in some weighted Sobolev spaces on unbounded domains.

1. Introduction

Let be an open subset of . The uniformly elliptic second-order linear differential operator with leading coefficients , and the associated Dirichlet problem have been extensively studied under different hypotheses of discontinuity on the coefficients of (we refer to [1] for a general survey on the subject). In particular, some bounds and the related existence and uniqueness results have been obtained.

Among the various hypotheses, in the framework of discontinuous coefficients, we are interested here in those of Miranda's type, having in mind the classical result of [2] where the leading coefficients have derivatives . First generalizations in this direction have been carried on, always considering a bounded and sufficiently regular set , assuming that the derivatives belong to some wider spaces. In particular, in [3], the are in the weak- space, while, in [4], they are supposed to be in an appropriate subspace of the classical Morrey space , where . In [5], the leading coefficients are supposed to be close to functions whose derivatives are in . A further extension, to a very general case, has been proved in [6, 7], supposing that the are in , which means a kind of continuity in the average sense and not in the pointwise sense.

In this paper, we deal with unbounded domains and we impose hypotheses of Miranda's type on the leading coefficients, assuming that their derivatives belong to a suitable Morrey type space, which is a generalization to unbounded domains of the classical Morrey space. The existence of the derivatives is of crucial relevance in our analysis, since it allows us to rewrite the operator in divergence form and puts us in position to use some known results concerning variational operators. A straightforward consequence of our argument is the following -bound, having the only term in the right-hand side, where the dependence of the constant is explicitly described (see Section 4). This kind of estimate often cannot be obtained when dealing with unbounded domains and clearly immediately takes to the uniqueness of the solution of problem (1.2).

In the framework of unbounded domains, under more regular boundary conditions, an analogous a priori bound can be found in [8], where different assumptions on the are taken into account. We quote here also the results of [9], where, in the spirit of [5], the leading coefficients are supposed to be close, in as specific sense, to functions whose derivatives are in spaces of Morrey type and have a suitable behavior at infinity.

The -bound obtained in (1.3) allows us to extend our result to a weighted case. The relevance of Sobolev spaces with weight in the study of the theory of PDEs with prescribed boundary conditions on unbounded open subsets of is well known. Indeed, in this framework, it is necessary to require not only conditions on the boundary of the set, but also conditions controlling the behaviour of the solution at infinity. In this order of ideas, we also consider the Dirichlet problem, where , , and are weighted Sobolev spaces where the weight is power of a function , of class , and such that see Sections 2 and 3 for more details. Also in this weighted case, we obtain the bound where the dependence of the constant is again completely determined. From this a priori estimate, in Section 5, we deduce the solvability of problem (1.4).

Existence and uniqueness results for similar problems in the weighted case, but with different weight functions and different assumptions on the coefficients, have been proved in [10]. Recent results concerning a priori estimates for solutions of the Poisson and heat equations in weighted spaces can be found in [11], where weights of Kondrat'ev type are considered.

2. A Class of Weighted Sobolev Spaces

Let be an open subset of , not necessarily bounded, . We want to introduce a class of weight functions defined on .

To this aim, given , we consider a function such that and As an example, we can think of the function In the following lemma, we show a property, needed in the sequel, concerning this class of weight functions.

Lemma 2.1. If assumption (2.1) is satisfied, then

Proof. The proof is obtained by induction. From (2.1), we get with positive constant depending only on . Thus (2.3) holds for .Now, let us assume that (2.3) holds for any such that and any , and fix a such that . Then, using (2.1) and by the induction hypothesis written for , we have with positive constant depending only on . Hence, (2.3) holds true also for .

Now, let us study some properties of a new class of weighted Sobolev spaces, with weight function of the above-mentioned type.

For , , and , and given a weight function satisfying (2.1), we define the space of distributions on such that equipped with the norm given in (2.6). Moreover, we denote by the closure of in and put .

Lemma 2.2. Let , , and . If assumption (2.1) is satisfied, then there exist two constants such that with and .

Proof. Observe that from (2.3), we have with depending only on . This entails the inequality on the right-hand side of (2.7).To get the left-hand side inequality, it is enough to show that with depending only on .We will prove (2.9) by induction. From (2.3), one has for , with depending only on . Hence, (2.9) holds for .If (2.3) holds for any such that , then, using again (2.3) and by the induction hypothesis, we have with depending only on .

Let us specify a density result.

Lemma 2.3. Let , , and . If has the segment property and assumption (2.1) is satisfied, then is dense in .

Proof. The proof follows by Lemma 2.2 in [12], since clearly both .

This allows us to prove the following inclusion.

Lemma 2.4. Let , , and . If has the segment property and assumption (2.1) is satisfied, then

Proof. The density result stated in Lemma 2.3 being true, we can argue as in the proof of Lemma 2.1 of [10] to obtain the claimed inclusion.

From this last lemma, we easily deduce that, if has the segment property, also .

Lemma 2.5. Let , , and . If has the segment property and assumption (2.1) is satisfied, then the map defines a topological isomorphism from to and from to .

Proof. The first part of the proof easily follows from Lemma 2.2 with . Let us show that if and only if .If , there exists a sequence converging to in . Therefore, fixed , there exists such that Fix , clearly , because of its compact support. Therefore, there exists a sequence converging to in . Hence, there exists such that Putting together (2.14) and (2.15), we get  . Thus, .Vice versa, if we assume that , we find a sequence converging to in . Hence, there exists such that Fix , since , which is contained in by Lemma 2.4, there exists a sequence converging to in . Therefore, there exists such that From (2.17) and (2.18), we get   , so that .

3. Preliminary Results

From now on, we consider a weight , , and such that (2.1) is satisfied (for ). Moreover, we assume that An example of a function verifying our hypotheses is given by We associate to a function defined by Clearly verifies (2.1) and

Now, let us fix a cutoff function such that Then, set By our definition, it follows that and

Finally, we introduce the sequence

For any , one has where depends only on . This entails that Concerning the derivatives, easy calculations give that, for any , with and positive constants independent of and .

From (3.9), (3.11), (3.13), (3.14), and (3.15), we obtain, for any , where and are positive constants independent of and .

Combining (3.13) and (3.16), we have also, for any ,

We conclude this section proving the following lemma.

Lemma 3.1. Let and , be defined by (3.3) and (3.6), respectively. Then

Proof. Set By the second relation in (3.4), the supremum of over is actually a maximum; thus, for every , there exists such that .To prove (3.19), we have to show that .We proceed by contradiction. Hence, let us assume that there exists such that, for any , there exists such that .If the sequence is bounded, there exists a subsequence converging to a limit , and by the continuity of , converges to . On the other hand, , thus , which is in contrast with the fact that is a convergent sequence.Therefore, is unbounded, so that there exists a subsequence such that . Thus, by the second relation in (3.4), one has . This gives the contradiction since .

4. A No Weighted A Priori Bound

We want to prove a bound for an uniformly elliptic second-order linear differential operator. Let us start recalling the definitions of the function spaces in which the coefficients of our operator will be chosen.

For any Lebesgue measurable subset of , let be the -algebra of all Lebesgue measurable subsets of . Given , we denote by the Lebesgue measure of , by its characteristic function, and by the intersection (), where is the open ball with center and radius .

For , , , and fixed in , the space of Morrey type is the set of all functions in such that endowed with the norm defined in (4.1). It is easily seen that, for any , a function belongs to if and only if it belongs to ; moreover, the norms of in these two spaces are equivalent. This allows us to restrict our attention to the space .

We now introduce three subspaces of needed in the sequel. The set is made up of the functions such that while and denote the closures of and in , respectively. We point out that We put , , , and .

We want to define the moduli of continuity of functions belonging to or . To this aim, let us put, for and , Recall first that for a function the following characterization holds: while where denotes a function of class such that Thus, if is a function in , a modulus of continuity of in is a map such that While, if belongs to , a modulus of continuity of in is an application such that If has the property where is a positive constant independent of and , it is possible to consider the space () of functions such that where If , where we say that if for .

If belongs to , a modulus of continuity of in is function such that For more details on the above-defined function spaces, we refer to [8, 13–15].

Let us start proving a useful lemma.

Lemma 4.1. If has the uniform -regularity property and then .

Proof. For , the result can be found in [16], combining Lemma 4.1 and the argument in the proof of Lemma 4.2.Concerning , we firstly apply a known extension result, see [9, Corollary 2.2], stating that any function such that admits an extension such that .Then, we prove that for all and , there exists a constant such that Indeed, in view of the above considerations, if (4.16) holds true, one has that , so .Consider the function By Poincaré-Wirtinger inequality and Hölder inequality, one gets this gives (4.16).

For reader’s convenience, we recall here some results proved in [17], adapted to our needs.

Lemma 4.2. If is an open subset of having the cone property and , with and if , and and if , then is a bounded operator from to . Moreover, there exists a constant , such that with . Furthermore, if , then for any there exists a constant , such that with . If , with and , then the operator in (4.19) is bounded from to . Moreover, there exists a constant , such that with . Furthermore, if , then for any there exists a constant , such that with .

Proof. The proof easily follows from Theorem 3.2 and Corollary 3.3 of [17].

From now on, we assume that is an unbounded open subset of , with the uniform -regularity property.

We consider the differential operator with the following conditions on the coefficients: We explicitly observe that under the assumptions – the operator is bounded, as a consequence of Lemma 4.2.

We are now in position to prove the above-mentioned a priori estimate.

Theorem 4.3. Let be defined in (4.24). Under hypotheses –, there exists a constant such that with .

Proof. Let us put and fix . Lemma 4.1 being true, Lemma 3.1 of [18] (for ) and Theorem 5.1 of [17] (for ) apply, so that there exists a constant such that with . Therefore, On the other hand, from Lemma 4.2, one has with and .Furthermore, classical interpolation results give that there exists a constant such that with . Combining (4.28), (4.29) and (4.30) we conclude that there exists such that with .To show (4.25), it remains to estimate . To this aim let us rewrite our operator in divergence form in order to adapt to our framework some known results concerning operators in variational form. Following along the lines, the proofs of Theorem 4.3 of [19] (for ) and of Theorem 4.2 of [13] (for ), with opportune modifications—we explicitly observe that the continuity of the bilinear form associated to (4.32) in our case is an immediate consequence of Lemma 4.2—we obtain that with . Putting together (4.31) and (4.33), we obtain (4.25).

5. Uniqueness and Existence Results

This section is devoted to the proof of the solvability of a Dirichlet problem for a class of second-order linear elliptic equations in the weighted space . The -bound obtained in Theorem 4.3 allows us to show the following a priori estimate in the weighted case.

Theorem 5.1. Let be defined in (4.24). Under hypotheses –, there exists a constant such that with ,.

Proof. Fix . In the sequel, for sake of simplicity, we will write , for a fixed . Observe that satisfies (2.1), as a consequence of (3.16), so that Lemma 2.5 applies giving that . Therefore, in view of Theorem 4.3, there exists , such that with . Easy computations give Putting together (5.2) and (5.3), we deduce that where depends on the same parameters as and on . On the other hand, from Lemma 4.2 and (3.17), we get with .Combining (3.17), (3.18), (5.4), and (5.5), with simple calculations we obtain the bound where depends on the same parameters as and on .By Lemma 3.1, it follows that there exists such that Now, if we still denote by the function , from (5.6) and (5.7), we deduce that Then, by Lemma 2.2 and by (3.12), written for , we have with depending on the same parameters as and on .This last estimate being true for every , we also have The bounds in (5.9) and (5.10) together with the definition (3.3) of give estimate (5.1).

Lemma 5.2. The Dirichlet problem where is uniquely solvable.

Proof. Observe that is a solution of problem (5.11) if and only if is a solution of the problem Clearly, for any , then (5.13) is equivalent to the problem where Using Theorem 5.2 in [18] (for ), Theorem 4.3 of [20] (for ), (1.6) of [8], and the hypotheses on , we obtain that (5.15) is uniquely solvable and then problem (5.11) is uniquely solvable too.

Theorem 5.3. Let be defined in (4.24). Under hypotheses –, the problem is uniquely solvable.

Proof. For each , we put In view of Theorem 5.1, Thus, taking into account the result of Lemma 5.2 and using the method of continuity along a parameter (see, e.g., Theorem 5.2 of [21]), we obtain the claimed result.

References

A. Maugeri, D. K. Palagachev, and L. G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, J.Wiley-VCH, New York, NY, USA, 2000.

C. Miranda, “Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui,” Annali di Matematica Pura ed Applicata, vol. 63, no. 1, pp. 353–386, 1963.

A. Alvino and G. Trombetti, “Second order elliptic equations whose coefficients have their first derivatives weakly-,” Annali di Matematica Pura ed Applicata, vol. 138, no. 1, pp. 331–340, 1984.

F. Chiarenza and M. Franciosi, “A generalization of a theorem by C. Miranda,” Annali di Matematica Pura ed Applicata, vol. 161, pp. 285–297, 1992.

M. Chicco, “Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients,” Annali di Matematica Pura ed Applicata, vol. 92, no. 1, pp. 13–22, 1972.

F. Chiarenza, M. Frasca, and P. Longo, “Interior estimates for nondivergence elliptic equations with discontinuous coefficients,” Ricerche di Matematica, vol. 40, no. 1, pp. 149–168, 1991.

F. Chiarenza, M. Frasca, and P. Longo, “-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,” Transactions of the American Mathematical Society, vol. 336, no. 2, pp. 841–853, 1993.

M. Transirico and M. Troisi, “Equazioni ellittiche del secondo ordine di tipo non variazionale in aperti non limitati,” Annali di Matematica Pura ed Applicata, vol. 152, no. 1, pp. 209–226, 1988.

A. Canale, P. di Gironimo, and A. Vitolo, “Functions with derivatives in spaces of Morrey type and elliptic equations in unbounded domains,” Studia Mathematica, vol. 128, no. 3, pp. 199–218, 1998.

P. di Gironimo and M. Transirico, “Second order elliptic equations in weighted Sobolev spaces on unbounded domains,” Rendiconti della Accademia Nazionale delle Scienze, vol. 15, no. 10, pp. 163–176, 1991.

A. Kubica and W. M. Zajaczkowski, “A priori estimates in weighted spaces for solutions of the Poisson and heat equations,” Applicationes Mathematicae, vol. 34, no. 4, pp. 431–444, 2007.

L. Sgambati and M. Troisi, “Limitazioni a priori per una classe di problemi ellittici in domini non limitati,” Note di Matematica, vol. 1, no. 2, pp. 225–259, 1981.

M. Transirico, M. Troisi, and A. Vitolo, “Spaces of Morrey type and elliptic equations in divergence form on unbounded domains,” Unione Matematica Italiana, vol. 9, no. 1, pp. 153–174, 1995.

L. Caso, R. D'Ambrosio, and S. Monsurrò, “Some remarks on spaces of Morrey type,” Abstract and Applied Analysis, vol. 2010, Article ID 242079, 22 pages, 2010.

M. Transirico, M. Troisi, and A. Vitolo, “BMO spaces on domains of ,” Ricerche di Matematica, vol. 45, no. 2, pp. 355–378, 1996.

L. Caso, P. Cavaliere, and M. Transirico, “A priori bounds for elliptic equations,” Ricerche di Matematica, vol. 51, no. 2, pp. 381–396, 2002.

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.

P. Cavaliere and M. Transirico, “The Dirichlet problem for elliptic equations in unbounded domains of the plane,” Journal of Function Spaces and Applications, vol. 8, no. 1, pp. 47–58, 2008.

M. Transirico and M. Troisi, “Equazioni ellittiche del secondo ordine a coefficienti discontinui e di tipo variazionale in aperti non limitati,” Unione Matematica Italiana, vol. 2, no. 1, pp. 385–398, 1988.

L. Caso, P. Cavaliere, and M. Transirico, “An existence result for elliptic equations with VMO—coefficients,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1095–1102, 2007.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Germany, 2nd edition, 1983.