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

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 Æ satisfies a suitable differential inequality, then vÆ(x)=supBÆ(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.

Abstract licence: CC BY-NC-ND

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