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Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems

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Abstract

Let be a divergence form operator with Lipschitz continuous coefficients in a domain, and let be a continuous weak solution of in. In this paper, we show that if satisfies a suitable differential inequality, then is a subsolution of away from its zero set. We apply this result to prove regularity of Lipschitz free boundaries in two-phase problems.

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References

  1. 1.

    Caffarelli LA:A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are. Revista Matemática Iberoamericana 1987,3(2):139-162.

  2. 2.

    Feldman M: Regularity for nonisotropic two-phase problems with Lipschitz free boundaries. Differential and Integral Equations 1997,10(6):1171-1179.

  3. 3.

    Wang P-Y:Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. I. Lipschitz free boundaries are. Communications on Pure and Applied Mathematics 2000,53(7):799-810. 10.1002/(SICI)1097-0312(200007)53:7<799::AID-CPA1>3.0.CO;2-Q

  4. 4.

    Feldman M: Regularity of Lipschitz free boundaries in two-phase problems for fully nonlinear elliptic equations. Indiana University Mathematics Journal 2001,50(3):1171-1200.

  5. 5.

    Cerutti MC, Ferrari F, Salsa S:Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are. Archive for Rational Mechanics and Analysis 2004,171(3):329-348. 10.1007/s00205-003-0290-5

  6. 6.

    Ferrari F:Two-phase problems for a class of fully nonlinear elliptic operators. Lipschitz free boundaries are. American Journal of Mathematics 2006,128(3):541-571. 10.1353/ajm.2006.0023

  7. 7.

    Caffarelli LA: A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz. Communications on Pure and Applied Mathematics 1989,42(1):55-78. 10.1002/cpa.3160420105

  8. 8.

    Caffarelli LA, Fabes E, Mortola S, Salsa S: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana University Mathematics Journal 1981,30(4):621-640. 10.1512/iumj.1981.30.30049

  9. 9.

    Jerison DS, Kenig CE: Boundary behavior of harmonic functions in nontangentially accessible domains. Advances in Mathematics 1982,46(1):80-147. 10.1016/0001-8708(82)90055-X

  10. 10.

    Brelot M: Axiomatique des Fonctions Harmoniques, Séminaire de Mathématiques Supérieures - Été 1965. Les Presses de l'Université de Montréal, Quebec; 1966.

  11. 11.

    Hervé R-M: Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel. Annales de l'Institut Fourier. Université de Grenoble 1962, 12: 415-571.

  12. 12.

    Hervé R-M:Un principe du maximum pour les sous-solutions locales d'une équation uniformément elliptique de la forme. Annales de l'Institut Fourier. Université de Grenoble 1964,14(2):493-507. 10.5802/aif.185

  13. 13.

    Hervé R-M, Hervé M: Les fonctions surharmoniques associées à un opérateur elliptique du second ordre à coefficients discontinus. Annales de l'Institut Fourier. Université de Grenoble 1969,19(1):305-359. 10.5802/aif.320

  14. 14.

    Littman W, Stampacchia G, Weinberger HF: Regular points for elliptic equations with discontinuous coefficients. Annali della Scuola Normale Superiore di Pisa, Serie III 1963, 17: 43-77.

  15. 15.

    Hervé R-M:Quelques propriétés des fonctions surharmoniques associées à une équation uniformément elliptique de la form. Annales de l'Institut Fourier. Université de Grenoble 1965,15(2):215-223. 10.5802/aif.214

  16. 16.

    Caffarelli LA:A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1988,15(4):583-602 (1989).

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Correspondence to Fausto Ferrari.

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Ferrari, F., Salsa, S. Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems. Bound Value Probl 2007, 057049 (2006) doi:10.1155/2007/57049

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Weak Solution
  • Functional Equation