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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 650870, 7 pages
A Priori Bounds in and in for Solutions of Elliptic Equations
Dipartimento di Matematica, Università di Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy
Received 21 March 2013; Accepted 15 May 2013
Academic Editor: Rodrigo Lopez Pouso
Copyright © 2013 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.
We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. The main theorem consists in an -a priori bound, . Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given.
The aim of this work is to give an overview on some recent results dealing with the study of a certain kind of the Dirichlet problem in the framework of unbounded domains. To be more precise, given an unbounded open subset of , , we are concerned with the elliptic second-order linear differential operator in variational form with coefficients and with the associated Dirichlet problem As far as we know, were Bottaro and Marina the first to approach this kind of problem who proved, in , an existence and uniqueness theorem for the solution of problem (2), for , assuming that
The study was later on generalized in  weakening the hypothesis (4) by considering coefficients , , and satisfying (4) only locally and for . Further improvements have been achieved in , for , since the , , and are taken in suitable Morrey type spaces with lower summabilities.
In two recent works, [4, 5], considering a more regular set and supposing that the lower order terms coefficients are as in  for and as in  for , we prove that if , then there exists a constant , whose dependence is completely described, such that for any bounded solution of (2) and for every . This can be done taking into account two different sign hypotheses, namely, (5) and the less common
Successively, in , we deepen the study begun in [4, 5] showing that to a bounded datum it corresponds a bounded solution . This allows us to prove, by means of an approximation argument, that if belongs to , , then the solution is in too and verifies (7). Putting together the two preliminary -estimates, , obtained under the different sign assumptions and adding the further hypothesis that the are also symmetric, by means of a duality argument, we finally obtain (7) for , for each sign hypothesis, assuming no boundedness of the solution and for .
To conclude, we provide two applications of our final -bound, , recalling the results of [7, 8] where our estimate plays a fundamental role in the study of certain weighted and non-weighted non-variational problems with leading coefficients satisfying hypotheses of Miranda’s type (see ). The nodal point in this analysis is the existence of the derivatives of the leading coefficients that allows us to rewrite the involved operator in variational form and avail ourselves of the above-mentioned a priori bound.
Always in the framework of unbounded domains, the study of different variational problems can be found in [10, 11]. Quasilinear elliptic equations with quadratic growth have been considered in . In [13–15] a very general weighted case, with principal coefficients having vanishing mean oscillation, has been taken into account.
2. A Class of Spaces of Morrey Type
In this section we recall the definitions and the main properties of a certain class of spaces of Morrey type where the coefficients of our operators belong. These spaces generalize the classical notion of Morrey spaces to unbounded domains and were introduced for the first time in ; see also  for some details. Thus, from now on, let be an unbounded open subset of , . By we denote the -algebra of all Lebesgue measurable subsets of . For , is its characteristic function, its Lebesgue measure, and (), where is the open ball with center in and radius . The class of restrictions to of functions is . For , is the class of all functions such that for any .
For and , the space of Morrey type is made up of all the functions in such that equipped with the norm defined in (9).
The closures of and in are denoted by and , respectively.
The following inclusion holds true: Moreover,
We put , , and .
Now, let us define the moduli of continuity of functions belonging to or . For and , we set Given a function , the following characterizations hold: 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
We finally recall two results of [4, 7], obtained adapting to our needs a more general theorem proved in , providing the boundedness and some embedding estimates for the multiplication operator where the function belongs to suitable spaces of Morrey type.
Theorem 1. If , with and if , and and if , then the operator in (17) is bounded from to . Moreover, there exists a constant such that
Let and . If is an open subset of having the cone property and , with if , then the operator in (17) is bounded from to . Moreover, there exists a constant such that with .
If , with if , then the operator in (17) is bounded from to . Moreover, there exists a constant such that with .
3. The Variational Problem
Consider, in an unbounded open subset of , , the second-order linear differential operator in divergence form Assume that the leading coefficients satisfy the hypotheses For the lower order terms coefficients suppose that Furthermore, let one of the following sign assumptions hold true: or in the distributional sense on , with positive constant.
It is natural to associate to the bilinear form , and observe that, in view of Theorem 1, the form is continuous on and so the operator is continuous too.
Let us start collecting some preliminary results concerning the existence and uniqueness of the solution of problem (22), as well as some a priori estimates. For the case where assumptions – are taken into account and for , we refer to  while for details can be found in . If , , and hold true, the results are proved in the more recent .
Theorem 2. Under hypotheses – (or , , and ), problem (22) is uniquely solvable and its solution satisfies the estimate where is a constant depending on , .
The next step in our analysis is to achieve an -estimate, , for the solution of (22) (see Theorem 8). This requires some additional hypotheses on the regularity of the set and on the datum , and some preparatory results that essentially rely on the introduction of certain auxiliary functions , used for the first time by Bottaro and Marina in  and employed in the framework of Morrey type spaces in . Let us give their definition and recall some useful properties.
Let and , with . For each we set
Lemma 3. Let , , and . Then there exist and , with , such that set one has and with positive constant.
In order to prove a fundamental preliminary estimate, obtained for (see Theorem 7), we need to take products involving the above defined functions as test functions in the variational formulation of our problem (23). To be more precise, in the first set of hypotheses (–), the test functions needed are . The following result ensures that these functions effectively belong to .
Lemma 4. If has the uniform -regularity property, then for every and for any one has
Lemma 4, whose rather technical proof can be found in , is a generalization of a known result by Stampacchia (see , or  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.
Once achieved (31), always in , we could prove the next lemma. Let be the functions of Lemma 3 obtained in correspondence of a given , of and of a positive real number specified in the proof of Lemma 4.1 of . One has the following.
Lemma 5. Let be the bilinear form defined in (23). If has the uniform -regularity property, under hypotheses –, there exists a constant such that where depends on .
If we consider the second set of hypotheses (, , and ), the test functions required in (23) are the products , obtained in correspondence of a fixed , of and of a positive real number specified in the proof of Lemma 4.1 of . In this last case and if has the uniform -regularity property, a result of  applies giving that , for any , . Hence, in  we could show the result.
Lemma 6. Let be the bilinear form in (23). If has the uniform -regularity property, under hypotheses , , and , there exists a constant such that where depends on .
The two lemmas just stated put us in a position to prove the following preliminary -a priori estimate, , in both sets of hypotheses; see also [4, 5]. We stress that here we require that both the datum and the solution are bounded.
Theorem 7. Under hypotheses – or , , and and if has the uniform -regularity property, is in and the solution of (22) is in , then and where is a constant depending on , .
Proof. Fix . We provide two different proofs in the cases that hypotheses or hold true.
Let – be satisfied. We consider the functions , , obtained in correspondence of the solution and of and of as in Lemma 4.1 of . In view of (29) we get with .
Hence, (32) entails that with and .
From the linearity of , (29), and (30), we have then with .
Using this last inequality and Hölder inequality we conclude our proof, since
If , , and hold, we consider again the functions , , obtained in correspondence of the solution and of as in the previous case, and of as in Lemma 4.1 of . In this second case, easy computations together with (29) give with .
Thus, from (33), we deduce that with and .
Hence, by (28) and Hölder inequality we obtain This ends the proof, in view of (30).
Theorem 8. Assume that hypotheses – or , , and are satisfied. If the set has the uniform -regularity property and the datum , for some , then the solution of problem (22) is in and where is a constant depending on , .
The proof, which is different according to hypothesis or , is essentially performed into two steps. In the first step, we show some regularity results, exploiting a technique introduced by Miranda in . Namely, we prove that if is the solution of (22) with , then, the datum being more regular, one also has . Thus Theorem 7 applies giving that and satisfies (34). The second step consists in considering a datum and then one can conclude by means of some approximation arguments; see also .
Finally, in , we prove the main result, that is, the claimed -bound, . To this aim, a further assumption on the leading coefficients is required: Then one has the following.
Theorem 9. Assume that hypotheses – or , , and are satisfied. If the set has the uniform -regularity property and the datum , for some , then the solution of problem (22) is in and where is a constant depending on , .
Proof. For , Theorems 2 and 8 already prove the result. It remains to show it for .
We assume that hypotheses – hold true. Under hypotheses , , and , a similar argument, with suitable modifications, can be used (we refer the reader to  for the details).
Let us define the bilinear form By one has Now consider the problem where, since , one gets .
As a consequence of Theorem 2 (in the second set of hypotheses) the solution of (46) exists and is unique. Furthermore, by Theorem 8 (in the second set of hypotheses) one also has Hence, if we denote by the solution of which exists and is unique in view of Theorem 2 (in the first set of hypotheses), we obtain Finally, taking in (49), we get the claimed result.
4. Non-Variational Problems
In this section, we show two applications of our main estimate (43).
To this aim, let and assume that Consider, then, the non-variational differential operator with the following conditions on the leading coefficients: Suppose that the lower order terms are such that In view of Theorem 1, under the assumptions –, the operator is bounded.
Theorem 10. Let be defined in (50). If hypotheses – are satisfied, then there exists a constant such that
with , .
Moreover, the problem is uniquely solvable.
The nodal point in achieving these results consists in the existence of the derivatives of the . Indeed, this consents to rewrite the operator in divergence form and exploit (43) in order to obtain an estimate as that in (51) but for more regular functions. Then, one can prove (51) by means of an approximation argument. Estimate (51) immediately takes to the solvability of problem (52) via a straightforward application of the method of continuity along a parameter, see, for instance, , and by the already known solvability of an opportune auxiliary problem.
As second application of (43), we obtain, in , an analogous of Theorem 10, in a weighted framework. Namely, we consider a weight function that is a power of a function of class such that and For instance, one can think of as the function
For , and , and given satisfying (53), we define the weighted Sobolev space as the space of distributions on such that endowed with the norm in (55). Furthermore, we denote the closure of in by and put .
In Theorems 4.2 and 5.2 of  we showed the following.
Theorem 11. Let be defined in (50). If hypotheses – are satisfied, then there exists a constant such that
with , .
Moreover, the problem
is uniquely solvable.
One of the main tools in the proof of Theorem 11 is given by the existence of a topological isomorphism from to and from to . This isomorphism consents to deduce by the non-weighted bound in (51) the corresponding weighted estimate in (56), taking into account also the imbedding results of Theorem 1. The existence and uniqueness of the solution of problem (57) follow then, as in the previous case, from a direct application of the method of continuity along a parameter by the solvability of a suitable auxiliary problem.
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