Boundary Value Problems
Volume 2007 (2007), Article ID 57049, 21 pages
doi:10.1155/2007/57049
Research Article

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

Fausto Ferrari1,2 and Sandro Salsa3

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

Received 29 May 2006; Accepted 10 September 2006

Academic Editor: José Miguel Urbano

Copyright © 2007 Fausto Ferrari and Sandro Salsa. 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.

Abstract

Let L be a divergence form operator with Lipschitz continuous coefficients in a domain Ω, and let u be a continuous weak solution of Lu=0 in {u≠0}. In this paper, we show that if φ satisfies a suitable differential inequality, then vφ(x)=supBφ(x)(x)u is a subsolution of Lu=0 away from its zero set. We apply this result to prove C1,γ regularity of Lipschitz free boundaries in two-phase problems.