Fully nonlinear free transmission problems

Abstract

We examine a free transmission problem driven by fully nonlinear elliptic operators. Since the transmission interface is determined endogenously, our analysis regards this object as a free boundary. We start by relating our problem with a pair of viscosity inequalities. Then, approximation methods ensure that strong solutions are of class $C^{1,Log-Lip}$, locally. In addition, under further conditions on the problem, we prove quadratic growth of the solutions away from branch points.