Skip to content

Advertisement

Boundary Value Problems
Boundary Value Problems Cover Image
  • Research Article
  • Open Access

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

Boundary Value Problems20062007:057049

https://doi.org/10.1155/2007/57049

©  Ferrari and Salsa 2007

  • Received: 29 May 2006
  • Accepted: 10 September 2006
  • Published:

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

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

[12345678910111213141516]

Authors’ Affiliations

(1)
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S.~Donato 5, Bologna, 40126, Italy
(2)
C.I.R.A.M., Via Saragozza 8, Bologna, 40123, Italy
(3)
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 7, Milano, 20133, Italy

References

  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.MATHMathSciNetView ArticleGoogle Scholar
  2. Feldman M: Regularity for nonisotropic two-phase problems with Lipschitz free boundaries. Differential and Integral Equations 1997,10(6):1171-1179.MATHMathSciNetGoogle Scholar
  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-QMATHMathSciNetView ArticleGoogle Scholar
  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.MATHMathSciNetView ArticleGoogle Scholar
  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/s00205-003-0290-5MATHMathSciNetView ArticleGoogle Scholar
  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.2006.0023MATHMathSciNetView ArticleGoogle Scholar
  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.3160420105MATHMathSciNetView ArticleGoogle Scholar
  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.1981.30.30049MATHMathSciNetView ArticleGoogle Scholar
  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-XMATHMathSciNetView ArticleGoogle Scholar
  10. Brelot M: Axiomatique des Fonctions Harmoniques, Séminaire de Mathématiques Supérieures - Été 1965. Les Presses de l'Université de Montréal, Quebec; 1966.Google Scholar
  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.MATHView ArticleGoogle Scholar
  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.185MATHView ArticleGoogle Scholar
  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.320MATHView ArticleGoogle Scholar
  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.MATHMathSciNetGoogle Scholar
  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.214MATHView ArticleGoogle Scholar
  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).MATHMathSciNetGoogle Scholar

Copyright

© Ferrari and Salsa 2007

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.

Advertisement