Free Boundary Regularity for Non-Convex Fully Nonlinear Alt-Phillips Problems

Alvis Zahl

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arXiv · 2608.01508

Free Boundary Regularity for Non-Convex Fully Nonlinear Alt-Phillips Problems

Alvis Zahl

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Abstract

This paper studies the fully nonlinear Alt-Phillips free boundary problem \begin{equation} F(D^2u) = u^\gamma \chi_{\{u>0\}}, \end{equation} for $\gamma \in (-1/3,1)$ without any convexity assumption on $F$. We establish that flat free boundaries are smooth. This result is new even for the fully nonlinear obstacle problem when $\gamma = 0$. Our approach is based on a partial hodograph transform, which converts the free boundary into a fixed boundary and produces a fully nonlinear degenerate equation. For this equation we prove a Harnack inequality, yielding an improvement of flatness, and a Schauder estimate for the associated linearized equation. Both estimates appear to be new and are of independent interest.

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Alvis Zahl. 2026-08-02. Free Boundary Regularity for Non-Convex Fully Nonlinear Alt-Phillips Problems. https://arxiv.org/abs/2608.01508

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