Boundary regularity for a fully nonlinear free transmission problem

TL;DR

This paper investigates boundary regularity in a fully nonlinear free transmission problem, establishing regularity estimates in Sobolev and $C^{1,\text{Log-Lip}}$ spaces by approximation methods, extending recent literature to free boundary scenarios.

Contribution

It introduces new regularity estimates for free transmission problems using approximation techniques, broadening the understanding of boundary behavior in nonlinear free boundary problems.

Findings

Established Sobolev and $C^{1,\text{Log-Lip}}$ regularity estimates

Extended recent results to free boundary settings

Used approximation methods comparing operators with limiting profiles

Abstract

We examine boundary regularity for a fully nonlinear free transmission problem. We argue using approximation methods, comparing the operators driving the problem with a limiting profile. Working natural conditions on the data of the problem, we produce regularity estimates in Sobolev and $C^{1,{\rm Log-Lip}}$-spaces. Our findings extend recent developments in the literature to the free boundary setting.