Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems
- Fausto Ferrari1, 2Email author and
- Sandro Salsa3
Received: 29 May 2006
Accepted: 10 September 2006
Published: 5 December 2006
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.
Keywords
Authors’ Affiliations
References
- 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.MATHMathSciNetView ArticleGoogle Scholar - Feldman M: Regularity for nonisotropic two-phase problems with Lipschitz free boundaries. Differential and Integral Equations 1997,10(6):1171-1179.MATHMathSciNetGoogle Scholar
- 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-QMATHMathSciNetView ArticleGoogle Scholar - 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.MATHMathSciNetView ArticleGoogle Scholar
- 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/s00205-003-0290-5MATHMathSciNetView ArticleGoogle Scholar - 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.2006.0023MATHMathSciNetView ArticleGoogle Scholar - 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.3160420105MATHMathSciNetView ArticleGoogle Scholar
- 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.1981.30.30049MATHMathSciNetView ArticleGoogle Scholar
- 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-XMATHMathSciNetView ArticleGoogle Scholar
- Brelot M: Axiomatique des Fonctions Harmoniques, Séminaire de Mathématiques Supérieures - Été 1965. Les Presses de l'Université de Montréal, Quebec; 1966.Google Scholar
- 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.MATHView ArticleGoogle Scholar
- 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.185MATHView ArticleGoogle Scholar - 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.320MATHView ArticleGoogle Scholar
- 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.MATHMathSciNetGoogle Scholar
- 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.214MATHView ArticleGoogle Scholar - 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).MATHMathSciNetGoogle Scholar
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