Keywords:elliptic equations; discontinuous coefficients; a priori bounds
We are interested in the Dirichlet problem
Considering unbounded domains, as far as we know, the first work on this subject goes back to , where Bottaro and Marina provide, for , an existence and uniqueness result for the solution of problem (1.1) assuming that
In this order of ideas, various generalizations have been performed still maintaining hypotheses (1.3) and (1.5) but weakening the condition (1.4). Indeed in , where the case is considered, and c are supposed to satisfy assumptions as those in (1.4), but just locally. Successively in , for , further improvements have been carried on since and c are in suitable Morrey-type spaces with lower summabilities.
where the dependence of the constant C on the data of the problem is fully determined.
More recently, in , supposing that the coefficients of lower-order terms are as in  for and as in  for , we showed that, for a sufficiently regular set Ω, and if , then there exists a constant C, whose dependence is completely described, such that
Here, in the same framework but replacing the classical hypothesis of sign (1.5) by the less common one
we establish two kinds of results for the solution of (1.1). First of all, we provide an existence and uniqueness theorem, then, taking into account an additional assumption on the regularity of the boundary of Ω, we prove the analogue of (1.7).
Let us briefly survey the way these results are achieved. In Section 2, we introduce the tools needed in the sequel. The definitions and some features of the Morrey-type spaces are given and some functions , related somehow to the solution of the problem and to the coefficients of the operator, are described, together with some specific properties. Section 3 is devoted to the solvability of problem (1.1). We start proving, by means of the above mentioned functions , the estimate in (1.6) that leads also to the uniqueness at once. Then, in view of well-known results of the operator theory, we get the existence verifying that L is a Fredholm operator with zero index. In the last section, we prove the claimed -estimate. This is done by means of a technical lemma, exploiting again the functions , which allows us to conclude.
We believe that the two estimates (1.7), obtained under the different sign assumptions, combined together should permit to prove, by means of a duality argument, that (1.7) holds true actually for any , considering one of the hypotheses (1.5) or (1.8) at a time.
For further studies of the Dirichlet problem for linear elliptic second order differential equations with discontinuous coefficients in divergence form in unbounded domains we refer the reader also to [11-13].
This section is devoted to the definitions and to some fundamental properties of the Morrey-type spaces where the coefficients of lower-order terms of our operator belong, and of some functions related to the solution of the problem and to all the coefficients of the operator (see the proofs of Theorem 3.1 and Lemma 4.1 for more details on this aspect) that are indispensable tools in the sequel.
Given an unbounded open subset Ω of , , we denote by the σ-algebra of all Lebesgue measurable subsets of Ω. For any , is its characteristic function and is the intersection (, ), where is the open ball centered in x and with radius r.
endowed with the norm above defined. Moreover, denotes the closure of in . These functional spaces generalize the classical notion of Morrey spaces to the case of unbounded domains and were introduced in  (we refer also to  where further characteristics are considered).
where the function g belongs to suitable spaces of Morrey type.
Proof The proofs of the properties (2.6), (2.7), (2.9), (2.11) and (2.12) can be found in .
Inequality (2.8) is an immediate consequence of (2.7).
This, together with (2.3) and (2.4), gives
On the other hand, by definition,
Combining (2.14) and (2.13), we conclude that
3 Existence and uniqueness result
We are interested in the study of the following Dirichlet problem in Ω:
where L is a second order linear differential operator in divergence form
satisfying the following hypotheses on the leading coefficients: Considering the coefficients of lower-order terms, we suppose that We associate to L the bilinear form
Proof We start proving estimate (3.4) that yields also to the uniqueness of the solution at once. Successively, in view of classical results concerning operator theory, to get the existence, it will be enough to verify that L is a Fredholm operator with zero index.
by (3.5) we get
Therefore, taking into account (2.6), we conclude that
This, together with (2.12), ends the proof of the bound in (3.4).
Now, as it was already mentioned, it only remains to show that the operator
is a Fredholm operator with zero index.
which is well defined in view of Lemma 2.1.
Then, consider the problem
Clearly, if we show that (3.6) has a unique solution, we end our proof, since in this case the operator L can be seen as a sum between a Fredholm operator with zero index and a compact operator; and therefore, it is a Fredholm operator with zero index itself.
Indeed, we explicitly observe that the operator
and the bounded one
To get the existence and uniqueness of the solution of problem (3.6), we want to make use of Lax-Milgram Lemma. Thus let us consider the bilinear form associated to it
On the other hand, Hölder and Young inequalities give that
This concludes the proof of Theorem 3.1. □
To this aim, we require a further assumption on the boundary of Ω:
Moreover, a technical lemma below is needed. We note that the proof of Lemma 4.1 follows the idea of the one of the estimate (3.4). However, in this case, there are some specific arguments that need to be explicitly treated.
whereCdepends ons, r, ν, μ.
Proof Let ugε and , for , be as above specified. Since , by definition of and by Lemma 2.2, the functions . Therefore, in view of hypothesis (), Lemma 3.2 in  applies giving that for any .
If we set
Now, we observe that explicit calculations give
Hence, putting together (4.3), (4.4) and (4.5), we get
Thus, by Young inequality,
Thus, by (4.1), one has
Hence by (2.8) and Hölder inequality, we get
This concludes the proof, in view of (2.12). □
The authors declare that they have no competing interests.
The authors conceived and wrote this article in collaboration and with the same responsibility. Both of them read and approved the final manuscript.
The authors would like to thank anonymous referees for a careful reading of this article and for valuable suggestions and comments.
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