Open Access

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

Boundary Value Problems20062007:057049

DOI: 10.1155/2007/57049

Received: 29 May 2006

Accepted: 10 September 2006

Published: 5 December 2006

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2F2007%2F57049/MediaObjects/13661_2006_Article_652_IEq1_HTML.gif be a divergence form operator with Lipschitz continuous coefficients in a domain https://static-content.springer.com/image/art%3A10.1155%2F2007%2F57049/MediaObjects/13661_2006_Article_652_IEq2_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2007%2F57049/MediaObjects/13661_2006_Article_652_IEq3_HTML.gif be a continuous weak solution of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F57049/MediaObjects/13661_2006_Article_652_IEq4_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2007%2F57049/MediaObjects/13661_2006_Article_652_IEq5_HTML.gif . In this paper, we show that if https://static-content.springer.com/image/art%3A10.1155%2F2007%2F57049/MediaObjects/13661_2006_Article_652_IEq6_HTML.gif satisfies a suitable differential inequality, then https://static-content.springer.com/image/art%3A10.1155%2F2007%2F57049/MediaObjects/13661_2006_Article_652_IEq7_HTML.gif is a subsolution of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F57049/MediaObjects/13661_2006_Article_652_IEq8_HTML.gif away from its zero set. We apply this result to prove https://static-content.springer.com/image/art%3A10.1155%2F2007%2F57049/MediaObjects/13661_2006_Article_652_IEq9_HTML.gif regularity of Lipschitz free boundaries in two-phase problems.

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Authors’ Affiliations

(1)
Dipartimento di Matematica, Università di Bologna
(2)
C.I.R.A.M.
(3)
Dipartimento di Matematica, Politecnico di Milano

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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.