Fully nonlinear free boundary problems: optimal boundary regularity beyond convexity

TL;DR

This paper proves optimal regularity for solutions to a broad class of fully nonlinear free boundary problems without convexity assumptions, using BMO estimates and differentiability conditions.

Contribution

It establishes the first optimal $C^{1,1}$ regularity results for fully nonlinear free boundary problems without convexity constraints.

Findings

Optimal $C^{1,1}$ regularity at boundary intersections

BMO estimates near fixed boundary

No convexity assumptions needed

Abstract

We study a general class of elliptic free boundary problems equipped with a Dirichlet boundary condition. Our primary result establishes an optimal $C^{1,1}$-regularity estimate for $L^p$-strong solutions at points where the free and fixed boundaries intersect. A key novelty is that no convexity or concavity assumptions are imposed on the fully nonlinear operator governing the system. Our analysis derives BMO estimates in a universal neighbourhood of the fixed boundary. It relies solely on a differentiability assumption. Once those estimates are available, applying by now standard methods yields the optimal regularity.