- Open Access
A weighted -a priori bound for a class of elliptic operators
© Monsurrò and Transirico; licensee Springer 2013
- Received: 28 February 2013
- Accepted: 7 May 2013
- Published: 27 May 2013
We prove a weighted -a priori bound, , for a class of uniformly elliptic second-order differential operators on unbounded domains. We deduce a uniqueness and existence result for the solution of the related Dirichlet problem.
MSC: 35J25, 35B45, 35R05.
- elliptic equations
- discontinuous coefficients
- a priori bounds
- weighted Sobolev spaces
In the last years the interest in weighted Sobolev spaces has been rapidly increasing, both for what concerns the structure of the spaces themselves as well as for their suitability in the theory of PDEs with prescribed boundary conditions. In particular, these spaces find a natural field of application in the case of unbounded domains. Indeed, in this setting, there is the need to impose not only conditions on the boundary of the set, but also conditions that control the behavior of the solution at infinity.
Among the various hypotheses involving discontinuous coefficients, we consider here those of Miranda type, referring to the classical paper  where the have derivatives in . Namely, we suppose that the belong to suitable Morrey-type spaces that extend to unbounded domains the classical notion of Morrey spaces (see Section 3 for the details).
Always in the framework of unbounded domains, no-weighted problems weakening the hypotheses of  have been studied. We quote here, for instance, [3–6] for and  for . A very general case, where the have vanishing mean oscillation (VMO), has been taken into account in  (for the pioneer works considering VMO assumptions in the framework of bounded domains, we refer to [9–11]) and in [12, 13] in a weighted contest. Variational problems can be found in [14–16]. Quasilinear elliptic equations with quadratic growth have been considered in .
where the dependence of the constant c is completely described. Estimate (1.2) allows us to deduce the solvability of the related Dirichlet problem. This work generalizes to all a previous result of  where a no-weighted and a weighted case have been analyzed for . In  we considered an analogous problem, with , but without weight. Related variational results were studied in [19–21].
We start recalling the definitions of our specific weight functions. Then we will focus on certain classes of related weighted Sobolev spaces recently introduced in , where a detailed description and the proofs of all the properties below can be found.
equipped with the norm given in (2.2). Furthermore, we denote the closure of in by , and put .
Let us quote the following fundamental tool, which will allow us to exploit no-weighted results in order to pass to the weighted case.
with and .
defines a topological isomorphism from to and from to .
Here we recall the definitions and the main properties of a class of spaces of Morrey type where the coefficients of our operator will be chosen. These spaces are a generalization to unbounded domains of the classical Morrey spaces and were introduced for the first time in ; see also  for more details.
Thus, in the sequel let Ω be an unbounded open subset of , . The σ-algebra of all Lebesgue measurable subsets of Ω is denoted by . Given , is its Lebesgue measure, its characteristic function and (, ), where is the open ball centered in x and with radius τ.
endowed with the norm defined in (3.1). One can easily check that, for any arbitrary , a function g belongs to if and only if it belongs to and the norms of g in these two spaces are equivalent. Hence, we limit our attention to the space .
The closures of and in are denoted by and , respectively.
We put , and .
(we refer the reader, for instance, to  for the definition).
Let us start observing that, in view of Lemma 3.1, under the assumptions ()-(), the operator is bounded.
Then let us recall some known results contained in Theorem 3.2 and Corollary 3.3 of .
is uniquely solvable.
Now we prove the claimed weighted -bound.
Proof Fix . If for , . Thus, in view of the isomorphism of Lemma 2.1, one has that .
where depends on the same parameters as and on s.
where depends on the same parameters as and on .
with depending on the same parameters as and on , that is (4.4).
This concludes the proof of Theorem 4.2. □
In this last section we exploit our weighted estimate in order to deduce the solvability of the related Dirichlet problem.
A preliminary result is needed.
if for .
with b given by (5.2).
Since σ verifies (2.1) and (2.6), by (1.6) of  (which gives that both and are in every , ) one has that we are in the hypotheses of Theorem 4.1, therefore (5.5) is uniquely solvable, and then problem (5.1) is uniquely solvable too.
with b given by (5.3).
The thesis follows then arguing as in the previous case. □
is uniquely solvable.
with b given by (5.2) if for , or by (5.3) if for .
Thus, taking into account the result of Lemma 5.1 and using the method of continuity along a parameter (see, e.g., Theorem 5.2 of ), we obtain the claimed result. □
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