A fully nonlinear problem with free boundary in the plane

TL;DR

This paper proves that bounded solutions to a specific fully nonlinear free boundary problem in the plane are necessarily one-dimensional, using maximum principle and convexity methods.

Contribution

It establishes a one-dimensionality result for bounded solutions to a fully nonlinear free boundary problem in the plane, a novel theoretical insight.

Findings

Bounded solutions are one-dimensional.

Uses maximum principle and convexity arguments.

Advances understanding of nonlinear free boundary problems.

Abstract

We prove that bounded solutions to an overdetermined fully nonlinear free boundary problem in the plane are one dimensional. Our proof relies on maximum principle techniques and convexity arguments.