Regularity for nonisotropic two-phase problems with Lipschitz free boundaries

Abstract

We study a 2-phase free boundary problem, in which the positive and negative parts of a solution satisfy two different elliptic equations, and a condition, involving normal derivatives from positive and negative sides holds on the free boundary in a weak sense. We show that if the free boundary is locally a graph of Lipschitz function, then it is $C^{1,\alpha} $ smooth. This is an extension of the result obtained by L.Caffarelli in the case when the positive and negative parts of a solution satisfy the same elliptic equation.