Open Access

Uniqueness results for elliptic problems with singular data

Boundary Value Problems20062006:98923

DOI: 10.1155/BVP/2006/98923

Received: 22 December 2005

Accepted: 12 June 2006

Published: 28 September 2006

Abstract

We obtain some uniqueness results for the Dirichlet problem for second-order elliptic equations in an unbounded open set without the cone property, and with data depending on appropriate weight functions. The leading coefficients of the elliptic operator are VMO functions. The hypotheses on the other coefficients involve the weight function.

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Authors’ Affiliations

(1)
Dipartimento di Matematica e Informatica, Facoltà di Scienze Matematiche, Fisiche e Naturali (MM. FF. NN.), Università degli Studi di Salerno

References

  1. Adams RA: Sobolev Spaces. Academic Press, New York; 1975:xviii+268.Google Scholar
  2. Caso L: Bounds for elliptic operators in weighted spaces. Journal of Inequalities and Applications 2006, 2006: 14 pages.MathSciNetView ArticleMATHGoogle Scholar
  3. Caso L: Regularity results for singular elliptic problems. Journal of Function Spaces and Applications 2006, 4: 17 pages.MathSciNetView ArticleMATHGoogle Scholar
  4. Caso L, Cavaliere P, Transirico M: Existence results for elliptic equations. Journal of Mathematical Analysis and Applications 2002,274(2):554–563. 10.1016/S0022-247X(02)00287-1MathSciNetView ArticleMATHGoogle Scholar
  5. Caso L, Cavaliere P, Transirico M: Solvability of the Dirichlet problem in for elliptic equations with discontinuous coefficients in unbounded domains. Le Matematiche (Catania) 2002,57(2):287–302 (2005).MathSciNetMATHGoogle Scholar
  6. 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.MathSciNetMATHGoogle Scholar
  7. Caso L, Transirico M: Some remarks on a class of weight functions. Commentationes Mathematicae Universitatis Carolinae 1996,37(3):469–477.MathSciNetMATHGoogle Scholar
  8. Caso L, Transirico M: The Dirichlet problem for second order elliptic equations with singular data. Acta Mathematica Hungarica 1997,76(1–2):1–16. 10.1007/BF02907048MathSciNetView ArticleMATHGoogle Scholar
  9. Cavaliere P, Transirico M, Troisi M: Uniqueness result for elliptic equations in unbounded domains. Le Matematiche (Catania) 1999,54(1):139–146 (2000).MathSciNetMATHGoogle Scholar
  10. Chiarenza F, Frasca M, Longo P: Interior estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche di Matematica 1991,40(1):149–168.MathSciNetMATHGoogle Scholar
  11. 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. 10.2307/2154379MathSciNetMATHGoogle Scholar
  12. Chicco M: Solvability of the Dirichlet problem in for a class of linear second order elliptic partial differential equations. Bollettino della Unione Matematica Italiana 1971, 4: 374–387.MathSciNetMATHGoogle Scholar
  13. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences. Volume 224. 2nd edition. Springer, Berlin; 1983:xiii+513.View ArticleMATHGoogle Scholar
  14. Greco D: Nuove formole integrali di maggiorazione per le soluzioni di un'equazione lineare di tipo ellittico ed applicazioni alla teoria del potenziale. Ricerche di Matematica 1956, 5: 126–149.MathSciNetMATHGoogle Scholar
  15. Košelev AI: On boundedness of of derivatives of solutions of elliptic differential equations. Matematicheskij Sbornik. Novaya Seriya 1956, 38(80): 359–372.MathSciNetGoogle Scholar
  16. Miranda C: Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui. Annali di Matematica Pura ed Applicata. Serie Quarta 1963,63(1):353–386. 10.1007/BF02412185MathSciNetView ArticleMATHGoogle Scholar
  17. Pucci C: Limitazioni per soluzioni di equazioni ellittiche. Annali di Matematica Pura ed Applicata. Serie Quarta 1966,74(1):15–30. 10.1007/BF02416445MathSciNetView ArticleMATHGoogle Scholar
  18. Transirico M, Troisi M: Second-order nonvariational elliptic equations in unbounded open sets. Annali di Matematica Pura ed Applicata. Serie Quarta 1988, 152: 209–226. 10.1007/BF01766150MathSciNetView ArticleMATHGoogle Scholar
  19. Transirico M, Troisi M: Further contributions to the study of second-order elliptic equations in unbounded open sets. Bollettino della Unione Matemàtica Italiana. Serie VII. B 1990,4(3):679–691.MathSciNetMATHGoogle Scholar
  20. Troisi M: On a class of weight functions. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni. Serie V. Parte I 1986,10(1):141–152.MathSciNetMATHGoogle Scholar
  21. Troisi M: On a class of weighted Sobolev spaces. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni. Serie V. Parte I 1986,10(1):177–189.MathSciNetMATHGoogle Scholar
  22. Vitanza C: A new contribution to the regularity for a class of elliptic second order equations with discontinuous coefficients. Le Matematiche (Catania) 1993,48(2):287–296 (1994).MathSciNetMATHGoogle Scholar

Copyright

© Loredana Caso 2006

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.