We study the Dirichlet problem for linear elliptic second order partial differential equations with discontinuous coefficients in divergence form in unbounded domains. We establish an existence and uniqueness result and we prove an a priori bound in , .
MSC: 35J25, 35B45, 35R05.
Keywords:elliptic equations; discontinuous coefficients; a priori bounds
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
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
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 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.
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 [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.
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  (we refer also to  where further characteristics are considered).
where the function g belongs to suitable spaces of Morrey type.
Lemma 2.1If , with and if , and and if , then the operator in (2.1) is bounded and there exists a constant such that
Moreover, if , then the operator in (2.1) is also compact.
For , we define the functions of the real variable t
Lemma 2.2Let , and . Then there exist and , with , such that set
one has and
with positive constant.
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).
Considering (2.10), observe that in the case it is a trivial consequence of (2.6).
This, together with (2.3) and (2.4), gives
On the other hand, by definition,
Combining (2.14) and (2.13), we conclude that
, . Hence by (2.6) we get (2.10). □
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 lower-order 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 well-known 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
by (3.5) we get
Thus, choosing we have
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 Lax-Milgram 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
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 constant such 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  applies giving that for any .
Thus, we can take as a test function in (3.3), obtaining by (2.9) that
If we set
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,
Choosing and we have
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
with . □
We are finally in position to prove the above mentioned -bound.
Theorem 4.2Assume that the hypotheses ( )-( ) are satisfied. Iffis in and 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
with and .
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.
Chicco, M, Venturino, M: Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain. Ann. Mat. Pura Appl.. 178, 325–338 (2000). Publisher Full Text
Lions, PL: Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat.. 78, 205–212 (1985)
Lions, PL: Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés II. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat.. 79, 178–183 (1985)