International Journal of Differential Equations

VolumeΒ 2011Β (2011), Article IDΒ 572824, 17 pages

http://dx.doi.org/10.1155/2011/572824

## A Priori Bounds for a Class of Elliptic Operators

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

Received 17 June 2011; Accepted 5 August 2011

Academic Editor: Charles E.Β Chidume

Copyright Β© 2011 Sara MonsurrΓ² 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
β. 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
β , 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, 13β15].

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 PoincarΓ©-Wirtinger inequality and HΓΆlder inequality, one gets
this gives (4.16).

For readerβs 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 β 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 β, 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 modificationsβwe explicitly observe that the continuity of the bilinear form associated to (4.32) in our case is an immediate consequence of Lemma 4.2βwe 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 β, 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 β, 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.

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