Abstract and Applied Analysis

Volume 2012 (2012), Article ID 376179, 15 pages

doi:10.1155/2012/376179

Research Article

An -Estimate for Weak Solutions of Elliptic Equations

Sara Monsurrò and Maria Transirico

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 [1–6]).

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 ([7–9]), 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

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