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

The Alt–Phillips problem interpolates between two of the most studied free boundary problems: the Bernoulli problem, and the obstacle problem. It provides a natural model for a broad class of semi-linear free boundary problems. 

In this talk, I will discuss the free boundary regularity for the fully nonlinear Alt–Phillips problem F(D^2u) = u^\gamma \chi_{{u>0}}, for \gamma \in (-1/3,1). Here F is smooth and uniformly elliptic, and in particular, we make no convexity assumption and establish that flat free boundaries are smooth. This result is new even for the obstacle case 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.