Abstract
Keywords:
elliptic equations; discontinuous coefficients; a priori bounds1 Introduction
We are interested in the Dirichlet problem
where Ω is an unbounded open subset of , and L is a linear uniformly elliptic second order differential operator with discontinuous coefficients in divergence form
If Ω is bounded, this problem is classical in literature and has been deeply analyzed taking into account various kinds of hypotheses on the coefficients (for more details see, for instance, [16]).
Considering unbounded domains, as far as we know, the first work on this subject goes back to [7], 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 [8], where the case is considered, and c are supposed to satisfy assumptions as those in (1.4), but just locally. Successively in [9], for , further improvements have been carried on since and c are in suitable Morreytype spaces with lower summabilities.
In [79] we also find the bound
where the dependence of the constant C on the data of the problem is fully determined.
More recently, in [10], supposing that the coefficients of lowerorder terms are as in [9] for and as in [8] 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
for any bounded solution u of (1.1) and for every .
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 Morreytype 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 wellknown 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.
Considering the case , we notice that, as a consequence of (1.6), the bound (1.7) is true under both sign hypotheses even supposing no regularity on the boundary of Ω.
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 [1113].
2 Tools
This section is devoted to the definitions and to some fundamental properties of the Morreytype spaces where the coefficients of lowerorder 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 σalgebra 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.
For and , the space of Morrey type is the set of all the functions g in such that
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 [9] (we refer also to [14] where further characteristics are considered).
For the reader’s convenience, in the next lemma we recall some results of [15] and [8,9] concerning the multiplication operator
where the function g belongs to suitable spaces of Morrey type.
Lemma 2.1If, withandif, andandif, then the operator in (2.1) is bounded and there exists a constantsuch that
Moreover, if, then the operator in (2.1) is also compact.
Now, let us deal with the above mentioned functions . They were employed for the first time in [7] and were studied in the framework of Morreytype spaces in [9].
For , we define the functions of the real variable t
and
Lemma 2.2Let, and. Then there existand, with, such that set
Proof The proofs of the properties (2.6), (2.7), (2.9), (2.11) and (2.12) can be found in [9].
Inequality (2.8) is an immediate consequence of (2.7).
Considering (2.10), observe that in the case it is a trivial consequence of (2.6).
Thus let us fix and such that . As already proved in [16] and in [7], in the case of unbounded domains, one has
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
Let Ω be an unbounded open subset of , .
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 lowerorder terms, we suppose that We associate to L the bilinear form
As a consequence of Lemma 2.1, a is continuous on ; and therefore, the operator is continuous too.
Theorem 3.1Under hypotheses ()(), problem (3.1) is uniquely solvable and its solutionusatisfies the estimate
whereCis a constant depending onn, t, ν, μ, , .
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.
Let , for , be the functions of Lemma 2.2 corresponding to a solution u of (3.1), to and to a positive real number ε that will be specified in the sequel.
By a wellknown characterization of the space , we have
Thus, if we take as a test function in the variational formulation of problem (3.1), by simple calculations and (2.9) and (2.10), we obtain
Hypotheses () and () together with (2.7) give then
On the other hand, by the Hölder inequality, the embedding results contained in Lemma 2.1 and using hypothesis () and (2.11), one has that there exists a constant such that
Hence, set
by (3.5) we get
If we rewrite the last inequality for and we estimate , then for and we estimate and so on, we get by substituting that
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.
To this aim, set and denote by γu, , the element of given by
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
is compact, since, by hypothesis () and Lemma 2.1, it is obtained as a composition between the compact operator
and the bounded one
where , , is the element of defined by
To get the existence and uniqueness of the solution of problem (3.6), we want to make use of LaxMilgram Lemma. Thus let us consider the bilinear form associated to it
The continuity of the form (3.7) can be easily obtained by Lemma 2.1. Considering the coercivity, for every , in view of hypotheses () and (), one has
On the other hand, Hölder and Young inequalities give that
and therefore,
This concludes the proof of Theorem 3.1. □
4 An a priori bound in
Here we want to prove, for a sufficiently regular datum f, a a priori estimate, , for a bounded solution of problem (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.
Let be the functions of Lemma 2.2 corresponding to a fixed , to and to a positive real number ε to be specified in the proof of Lemma 4.1. The following result holds true:
Lemma 4.1Letabe the bilinear form in (3.3). Under hypotheses ()(), there exists a constantsuch that
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 [17] applies giving that for any .
Thus, we can take as a test function in (3.3), obtaining by (2.9) that
If we set
and
by hypotheses () and () and in view of (2.7), one has
On the other hand, by (2.6), (2.8) and (2.10), using the Hölder inequality, we get that there exists a constant , such that
Thus, using hypothesis (), by Lemma 2.1 and (2.11), we obtain
Now, we observe that explicit calculations give
Hence, putting together (4.3), (4.4) and (4.5), we get
Thus, by Young inequality,
If we rewrite the last inequality for , then for and take into account the estimate of obtained in the previous step, and so on, we conclude our proof. Indeed, we get
We are finally in position to prove the above mentioned bound.
Theorem 4.2Assume that the hypotheses ()() are satisfied. Iffis inand the solutionuof (3.1) is in, then
whereCis a constant depending onn, t, p, ν, μ, , .
Proof Fix . If we consider the functions , , corresponding to the solution u, to g and ε as in Lemma 4.1, easy computations together with (2.6) give that
Thus, by (4.1), one has
Hence by (2.8) and Hölder inequality, we get
This concludes the proof, in view of (2.12). □
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
The authors conceived and wrote this article in collaboration and with the same responsibility. Both of them read and approved the final manuscript.
Acknowledgement
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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