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

Recommended by José Miguel Urbano

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.