A fully nonlinear free transmission problem

Abstract

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 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 C 1,1 -regular function, with universal estimates.

Existence of solutions to a fully nonlinear free transmission problem

“…Interior regularity of L d -strong solutions to (1), as well as a preliminary analysis of the free boundary, have been studied in [26]. Nonetheless, the issue of existence is not discussed in that paper and it remained open so far.…”