Boundary Value Problems
Volume 2007 (2007), Article ID 57049, 21 pages
doi:10.1155/2007/57049
Subsolutions of elliptic operators in divergence form and application to two-phase free boundary problems
Fausto Ferrari1
and Sandro Salsa2
1Dipartimento di Matematica, Università di Bologna, Piazza di Porta S.~Donato 5, Bologna 40126, Italy
2Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 7, Milano 20133, Italy
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.