A fully nonlinear free transmission problem

We examine a free transmission problem driven by fully nonlinear elliptic operators. Since the transmission interface is determined endogeneously, our analysis is two-fold: we study the regularity of the solutions and geometric properties of the free boundary. We prove that strong solutions are locally of class $\mathcal{C}^{1,1}$, locally. As regards the free boundary we start by establishing weak results, such as its non-degeneracy, and proceed with the characterization of global solutions. Then, we turn our attention to the set of non-degenerate points. We find this set inherits the regularity of the solutions. That is, it is locally the graph of a $\mathcal{C}^{1,1}$-regular function, with universal estimates.