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Uniqueness results for elliptic problems with singular data

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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References

  1. 1.

    Adams RA: Sobolev Spaces. Academic Press, New York; 1975:xviii+268.

    Google Scholar 

  2. 2.

    Caso L: Bounds for elliptic operators in weighted spaces. Journal of Inequalities and Applications 2006, 2006: 14 pages.

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Caso L: Regularity results for singular elliptic problems. Journal of Function Spaces and Applications 2006, 4: 17 pages.

    MathSciNet  Article  MATH  Google Scholar 

  4. 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)00287-1

    MathSciNet  Article  MATH  Google Scholar 

  5. 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).

    MathSciNet  MATH  Google Scholar 

  6. 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.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Caso L, Transirico M: Some remarks on a class of weight functions. Commentationes Mathematicae Universitatis Carolinae 1996,37(3):469–477.

    MathSciNet  MATH  Google Scholar 

  8. 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/BF02907048

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Cavaliere P, Transirico M, Troisi M: Uniqueness result for elliptic equations in unbounded domains. Le Matematiche (Catania) 1999,54(1):139–146 (2000).

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Chiarenza F, Frasca M, Longo P: Interior estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche di Matematica 1991,40(1):149–168.

    MathSciNet  MATH  Google Scholar 

  11. 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. 10.2307/2154379

    MathSciNet  MATH  Google Scholar 

  12. 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.

    MathSciNet  MATH  Google Scholar 

  13. 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.

    Google Scholar 

  14. 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.

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Košelev AI: On boundedness of of derivatives of solutions of elliptic differential equations. Matematicheskij Sbornik. Novaya Seriya 1956, 38(80): 359–372.

    MathSciNet  Google Scholar 

  16. 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/BF02412185

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Pucci C: Limitazioni per soluzioni di equazioni ellittiche. Annali di Matematica Pura ed Applicata. Serie Quarta 1966,74(1):15–30. 10.1007/BF02416445

    MathSciNet  Article  MATH  Google Scholar 

  18. 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/BF01766150

    MathSciNet  Article  MATH  Google Scholar 

  19. 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.

    MathSciNet  MATH  Google Scholar 

  20. 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.

    MathSciNet  MATH  Google Scholar 

  21. 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.

    MathSciNet  MATH  Google Scholar 

  22. 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).

    MathSciNet  MATH  Google Scholar 

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Correspondence to Loredana Caso.

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Caso, L. Uniqueness results for elliptic problems with singular data. Bound Value Probl 2006, 98923 (2006). https://doi.org/10.1155/BVP/2006/98923

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Weight Function
  • Functional Equation