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

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.