Existence of solutions to a fully nonlinear free transmission problem

TL;DR

This paper proves the existence of solutions for a complex free boundary problem governed by a discontinuous, fully nonlinear operator, using approximation and iterative methods under natural assumptions.

Contribution

It introduces a novel approach to establish solutions for a class of non-variational free boundary problems with discontinuous operators.

Findings

Existence of $L^p$-viscosity solutions.

Existence of strong solutions.

Applicable iterative scheme for non-variational problems.

Abstract

We study an equation governed by a discontinuous fully nonlinear operator. Such discontinuities are solution-dependent, which introduces a free boundary. Working under natural assumptions, we prove the existence of $L^p$-viscosity and strong solutions to the problem. The operator does not satisfy the usual structure conditions and to obtain the existence of solutions we resort to solving approximate problems, combined with an iterative scheme. We believe our strategy can be applied to other classes of non-variational free boundary problems.