Regularity theory for fully nonlinear parabolic obstacle problems

TL;DR

This paper investigates the structure of the free boundary in fully nonlinear parabolic obstacle problems, proving smoothness near regular points and describing the singular set as a Lipschitz manifold.

Contribution

It provides a detailed regularity analysis of the free boundary, distinguishing between regular and singular parts, and characterizes the singular set as a Lipschitz manifold.

Findings

Free boundary is $C^ abla$ in space and time near regular points.

Singular points form a Lipschitz manifold of dimension $n-1$.

The singular set is $ ext{ extbackslash epsilon}$-flat in space for any $ ext{ extbackslash epsilon}>0$.

Abstract

We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is $C^\infty$ in space and time. Furthermore, we prove that the set of singular points is locally covered by a Lipschitz manifold of dimension $n-1$ which is also $\varepsilon$-flat in space, for any $\varepsilon>0$.