Solvability of the Dirichlet Problem for Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains
© Serena Boccia et al. 2008
Received: 2 March 2008
Accepted: 14 August 2008
Published: 20 August 2008
This paper is concerned with the study of the Dirichlet problem for a class of second-order linear elliptic equations in weighted Sobolev spaces on unbounded domains of , . We state a regularity result and we can deduce an existence and uniqueness theorem.
with coefficients .
It is well known that if is bounded, the above problem has been largely studied by several authors under various hypotheses of discontinuity on the leading coefficients and considering the case . In particular, some -bounds for the solutions of the problem (1.1) and related existence and uniqueness theorems have been obtained. Among the other results on this subject, we quote here the classical result of , where the author assumed that the 's belong to . This result was later generalized in different ways, supposing that the derivatives of the leading coefficients belong to some wider spaces. More recently, a relevant contribution to the theory has been given in [2–5], where the coefficients are assumed to be in the class VMO and ; observe here that VMO contains the class .
If the set is unbounded, under assumptions similar to those required in , problem (1.1) has for instance been studied in  with , and in  with . Instead, in [8, 9], the leading coefficients satisfy restrictions similar to those in [2, 3].
where and are some weighted Sobolev spaces and the weight functions are a suitable power of . We obtained that the operator has closed range and that for the problem (1.4) a uniqueness result holds.
In this paper, we study again the problem (1.4). We state a regularity result which allows us to obtain the solvability of the problem.
A similar weighted case was studied in  with the leading coefficients satisfying hypotheses of Miranda's type and when .
2. Weight Functions and Weighted Spaces
Let be any Lebesgue measurable subset of and let be the collection of all Lebesgue measurable subsets of . Let . Denote by the Lebesgue measure of , by the characteristic function of and by the class of restrictions to of functions with . Moreover, if is a space of functions defined on , we denote by the class of all functions such that for any . Finally, for any and , we put , and
where is independent of and .
We note that the class of all functions which are Lipschitz continuous in with Lipschitz coefficient is contained in (see ).
We assign an unbounded open subset of .
and note that .
where depend only on (see ).
If is a real function defined in , we denote by the zero extension of in .
We begin to prove the following.
and (2.16) yields the inequality (2.10).
Moreover, denote by the closure of in and put . A more detailed account of properties of the above-defined spaces can be found, for instance, in .
From Lemma 2.1 we can deduce another lemma which we will need in the proof of our regularity result.
with dependent only on and .
The first part of the lemma follows from (2.9) for and , if one uses (2.1) and (2.8). The second part of the lemma follows in a similar way from the inequality (2.10), if one puts , , and .
3. An Embedding Lemma
We consider the following condition:
From [17, Theorem 3.1] we have the following.
If the assumption (h0) holds, then for any , it results that?? and
with dependent only on and
4. A Regularity Result
with the following conditions on the coefficients:
there exist functions , , and such that
Observe that under the assumptions (h1)–(h3), it follows that the operator is bounded from Lemma 3.1.
where and . Then belongs to .
where depends only on .
with depending on the same parameters of .
with dependent on and .
where depends on and the constants and depend on .
where depends on , , , , .
with dependent on the same parameters of .
where depends on the same parameters of .
with dependent on the same parameters of and on .
Therefore, from (4.21), we have the result.
5. Existence and Uniqueness Results
In this section, we will prove our existence and uniqueness theorem. To this aim, we need two preliminary lemmas.
where is independent of (see ).
The Dirichlet problem
is uniquely solvable. Moreover, if , then the solution belongs to for all in .
Moreover, if , then also . Therefore, using the theorem in , we have that the solution of (5.6) belongs to for all , and so the solution of (5.2) lies in for all .
is uniquely solvable, where is defined by (5.3).
Let be a function in . Then, by Lemma 5.1, there exists a unique (for all ) such that .
Firstly, suppose that . It follows from Theorem 4.1 that belongs to . Moreover, by [10, Lemma 2.2], lies in .
Suppose now . Then (for all ) and then, using again Theorem 4.1, belongs to . Moreover, by [10, Lemma 2.2], lies in .
Therefore, in both cases, and it is a solution of the equation , so that . Since is dense in (see [14, Proposition 1.1]) and is a closed subspace of by [10, Theorem 4.1], we obtain that . The uniqueness of the solution follows from [10, Theorem 5.2].
Finally, adding the following assumption on the coefficients of :
we are now in position to state the following uniqueness and existence result.
is uniquely solvable.
is uniquely solvable.
is likewise uniquely solvable. The proof is complete.
- Miranda C: Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui. Annali di Matematica Pura ed Applicata 1963,63(1):353-386. 10.1007/BF02412185MATHView ArticleGoogle Scholar
- Chiarenza F, Frasca M, Longo P:Interior estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche di Matematica 1991,40(1):149-168.MATHMathSciNetGoogle Scholar
- Chiarenza F, Frasca M, Longo P: -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Transactions of the American Mathematical Society 1993,336(2):841-853. 10.2307/2154379MATHMathSciNetGoogle Scholar
- Vitanza C: -regularity for a class of elliptic second order equations with discontinuous coefficients. Le Matematiche 1992,47(1):177-186.MATHMathSciNetGoogle Scholar
- Vitanza C:A new contribution to the regularity for a class of elliptic second order equations with discontinuous coefficients. Le Matematiche 1993,48(2):287-296.MATHMathSciNetGoogle Scholar
- Transirico M, Troisi M: Equazioni ellittiche del secondo ordine di tipo non variazionale in aperti non limitati. Annali di Matematica Pura ed Applicata 1988,152(1):209-226. 10.1007/BF01766150MATHMathSciNetView ArticleGoogle Scholar
- Caso L, Cavaliere P, Transirico M:Solvability of the Dirichlet problem in for elliptic equations with discontinuous coefficients in unbounded domains. Le Matematiche 2002,57(2):287-302.MATHMathSciNetGoogle Scholar
- Caso L, Cavaliere P, Transirico M: Uniqueness results for elliptic equations VMO-coefficients. International Journal of Pure and Applied Mathematics 2004,13(4):499-512.MATHMathSciNetGoogle Scholar
- Caso L, Cavaliere P, Transirico M: An existence result for elliptic equations with VMO-coefficients. Journal of Mathematical Analysis and Applications 2007,325(2):1095-1102. 10.1016/j.jmaa.2006.02.048MATHMathSciNetView ArticleGoogle Scholar
- Boccia S, Monsurrò S, Transirico M: Elliptic equations in weighted Sobolev spaces on unbounded domains. International Journal of Mathematics and Mathematical Sciences to appear
- Di Gironimo P, Transirico M: Second order elliptic equations in weighted Sobolev spaces on unbounded domains. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Serie V. Memorie di Matematica 1991, 15: 163-176.MATHMathSciNetGoogle Scholar
- Troisi M: Su una classe di funzioni peso. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Serie V. Memorie di Matematica 1986,10(1):141-152.MATHMathSciNetGoogle Scholar
- Caso L, Transirico M: Some remarks on a class of weight functions. Commentationes Mathematicae Universitatis Carolinae 1996,37(3):469-477.MATHMathSciNetGoogle Scholar
- Troisi M: Su una classe di spazi di Sobolev con peso. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Serie V. Memorie di Matematica 1986,10(1):177-189.MATHMathSciNetGoogle Scholar
- Transirico M, Troisi M, Vitolo A: Spaces of Morrey type and elliptic equations in divergence form on unbounded domains. Bollettino dell'Unione Matematica Italiana. Serie VIII. B 1995,9(1):153-174.MATHMathSciNetGoogle Scholar
- Transirico M, Troisi M, Vitolo A: spaces on domains of Ricerche di Matematica 1996,45(2):355-378.MATHMathSciNetGoogle Scholar
- Glushak AV, Transirico M, Troisi M: Teoremi di immersione ed equazioni ellittiche in aperti non limitati. Rendiconti di Matematica e delle sue Applicazioni 1989,9(1):113-130.MATHMathSciNetGoogle Scholar
- Boccia S, Caso L: Interpolation inequalities in weighted Sobolev spaces. Journal of Mathematical Inequalities to appear
- Miranda C: Teoremi di unicità in domini non limitati e teoremi di Liouville per le soluzioni dei problemi al contorno relativi alle equazioni ellittiche. Annali di Matematica Pura ed Applicata 1962,59(1):189-212. 10.1007/BF02411730MATHView ArticleGoogle Scholar
- Lions P-L: Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés. Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 1985,78(5):205-212.MATHMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.