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.
Keywordselliptic 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].
2 Weight functions and weighted spaces
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.
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 .
3 A class of spaces of Morrey type
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 .
4 An a priori bound
(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.
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. □
5 Some uniqueness and existence results
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. □
- Monsurrò S, Salvato M, Transirico M: A priori bounds for a class of elliptic operators . Int. J. Differ. Equ. 2011. doi:10.1155/2011/572824Google Scholar
- Miranda C: Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui . Ann. Mat. Pura Appl. 1963, 63: 353–386. 10.1007/BF02412185MathSciNetView ArticleMATHGoogle Scholar
- Transirico M, Troisi M: Equazioni ellittiche del secondo ordine di tipo non variazionale in aperti non limitati . Ann. Mat. Pura Appl. 1988, 152: 209–226. 10.1007/BF01766150MathSciNetView ArticleGoogle Scholar
- Transirico M, Troisi M: Ulteriori contributi allo studio delle equazioni ellittiche del secondo ordine in aperti non limitati . Boll. Unione Mat. Ital., B 1990, 4: 679–691.MathSciNetGoogle Scholar
- Caso L, Cavaliere P, Transirico M: A priori bounds for elliptic equations . Ric. Mat. 2002, 51: 381–396.MathSciNetMATHGoogle Scholar
- Caso L, Cavaliere P, Transirico M: Existence results for elliptic equations . J. Math. Anal. Appl. 2002, 274: 554–563. 10.1016/S0022-247X(02)00287-1MathSciNetView ArticleMATHGoogle Scholar
- Caso L, Cavaliere P, Transirico M:Solvability of the Dirichlet problem in for elliptic equations with discontinuous coefficients in unbounded domains . Matematiche 2002, 57: 287–302.MathSciNetMATHGoogle Scholar
- Caso L, Cavaliere P, Transirico M:An existence result for elliptic equations with -coefficients . J. Math. Anal. Appl. 2007, 325: 1095–1102. 10.1016/j.jmaa.2006.02.048MathSciNetView ArticleMATHGoogle Scholar
- Chiarenza F, Frasca M, Longo P:Interior estimates for non divergence elliptic equations with discontinuous coefficients . Ric. Mat. 1991, 40: 149–168.MathSciNetMATHGoogle Scholar
- Chiarenza F, Frasca M, Longo P:-Solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients . Trans. Am. Math. Soc. 1993, 336: 841–853.MathSciNetMATHGoogle Scholar
- Ragusa MA: Cauchy-Dirichlet problem associated to divergence form parabolic equations . Commun. Contemp. Math. 2004, 6(3):377–393. 10.1142/S0219199704001392MathSciNetView ArticleMATHGoogle Scholar
- Boccia S, Monsurrò S, Transirico M: Elliptic equations in weighted Sobolev spaces on unbounded domains . Int. J. Math. Math. Sci. 2008. doi:10.1155/2008/582435Google Scholar
- Boccia S, Monsurrò S, Transirico M: Solvability of the Dirichlet problem for elliptic equations in weighted Sobolev spaces on unbounded domains . Bound. Value Probl. 2008. doi:10.1155/2008/901503Google Scholar
- Bottaro G, Marina ME: Problema di Dirichlet per equazioni ellittiche di tipo varia-zionale su insiemi non limitati . Boll. Unione Mat. Ital. 1973, 8(4):46–56.MathSciNetMATHGoogle Scholar
- 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. 1985, 78(8):205–212.MATHGoogle Scholar
- 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. 1985, 79(8):178–183.MATHGoogle Scholar
- Donato P, Giachetti D: Quasilinear elliptic equations with quadratic growth on unbounded domains . Nonlinear Anal. 1986, 8(10):791–804.MathSciNetView ArticleMATHGoogle Scholar
- Monsurrò S, Transirico M:A -estimate for a class of elliptic operators . Int. J. Pure Appl. Math. 2013, 83(3):489–499. doi:10.12732/ijpam.v83i3.9View ArticleMATHGoogle Scholar
- Monsurrò S, Transirico M:A -estimate for weak solutions of elliptic equations . Abstr. Appl. Anal. 2012. doi:10.1155/2012/376179Google Scholar
- Monsurrò S, Transirico M: Dirichlet problem for divergence form elliptic equations with discontinuous coefficients . Bound. Value Probl. 2012. doi:10.1186/1687–2770–2012–67Google Scholar
- Monsurrò S, Transirico M:A priori bounds in for solutions of elliptic equations in divergence form . Bull. Sci. Math. 2013. (in press). doi:10.1016/j.bulsci.2013.02.002Google Scholar
- Transirico M, Troisi M, Vitolo A: Spaces of Morrey type and elliptic equations in divergence form on unbounded domains . Boll. Unione Mat. Ital., B 1995, 9(7):153–174.MathSciNetMATHGoogle Scholar
- Caso L, D’Ambrosio R, Monsurrò S: Some remarks on spaces of Morrey type . Abstr. Appl. Anal. 2010. doi:10.1155/2010/242079Google Scholar
- Cavaliere P, Longobardi M, Vitolo A: Imbedding estimates and elliptic equations with discontinuous coefficients in unbounded domains . Matematiche 1996, 51: 87–104.MathSciNetMATHGoogle Scholar
- Adams RA: Sobolev Spaces. Academic Press, New York; 1975.MATHGoogle Scholar
- Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. 2nd edition. Springer, Berlin; 1983.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.