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 , and let u be a continuous weak solution of Lu=0 in {u ￠ ‰ 0}. In this paper, we show that if satisfies a suitable differential inequality, then v (x)=supB (x)(x)u is a subsolution of Lu=0 away from its zero set. We apply this result to prove C1, 3 regularity of Lipschitz free boundaries in two-phase problems.