We study a free transmission problem driven by degenerate fully nonlinear operators. By framing the equation in the context of viscosity inequalities, we produce optimal regularity results for viscosity solutions and certain strong solutions to the problem. Our findings include regularity in C 1,α spaces, and an explicit characterization of α in terms of the degeneracy rates. As a consequence, we examine geometric properties of the associated free boundary. We argue by perturbation methods, relating our problem to a homogeneous, fully nonlinear uniformly elliptic equation.

The aim of the paper is to generalize the author's previous work [15]. We extend the argument [15] for any uniformly elliptic operator in divergence form Lu = −div(A(x)∇u), more precisely, we study a fractional type degenerate elliptic equation posed in bounded domains with homogeneous boundary conditionswhere L −s is the inverse s-fractional elliptic operator for any s ∈ (0, 1). This work consists of two part. The first part is devoted to state how the boundary condition will be consider (in the spirit of F. Otto [25]), and to give a formulation for the IBVP. In the second part, It is shown the existence of mass-preserving, non-negative weak solutions satisfying energy estimates for measurable and bounded non-negative initial data.

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