A fully nonlinear free transmission problem

TL;DR

This paper investigates a free transmission problem involving fully nonlinear elliptic operators, establishing regularity of solutions and geometric properties of the free boundary, with results on solution smoothness and boundary behavior.

Contribution

It introduces a new analysis framework for free transmission problems with fully nonlinear operators, proving local $C^{1,1}$ regularity and characterizing free boundary solutions.

Findings

Strong solutions are locally $C^{1,1}$

Free boundary is non-degenerate

Global solutions are characterized

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 some geometric properties of the free boundary. By relating our problem with a pair of viscosity inequalities, we prove that strong solutions are 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.