Two-phase problems: Perron solutions and regularity of the Neumann problem in convex cones

TL;DR

This paper extends interior regularity results for fully nonlinear two-phase free boundary problems with Neumann conditions to convex domains with corners, establishing existence and boundary regularity under mild assumptions.

Contribution

It introduces a boundary regularity extension of Caffarelli's classical two-phase problem theory to convex sets with corners, including a new existence theorem for Dirichlet problems.

Findings

Interior regularity extends up to the boundary in convex cones.

Established existence of solutions for Dirichlet two-phase problems with nonlinear operators.

Regularity results hold under mild convexity assumptions.

Abstract

We investigate a fully nonlinear two-phase free boundary problem with a Neumann boundary condition on the boundary of a general convex set $K \subset \mathbb{R}^n$ with corners. We show that the interior regularity theory developed by Caffarelli for the classical two-phase problem in his pioneer works \cite{C1,C2}, can be extended up to the boundary for the Neumann boundary condition under very mild regularity assumptions on the convex domain $K$. To start, we establish a general existence theorem for the Dirichlet two-phase problem driven by two different fully nonlinear operators, which is a result of independent interest.