Regularity of the Free Boundary for a Bernoulli-Type Parabolic Problem with Variable Coefficients

TL;DR

This paper proves that Lipschitz free boundaries in a parabolic Bernoulli-type problem with variable coefficients are actually smooth ($C^1$), with a normal vector that changes smoothly, ensuring classical solutions.

Contribution

It establishes the $C^1$ regularity of Lipschitz free boundaries for a class of parabolic free boundary problems with variable coefficients, improving understanding of boundary smoothness.

Findings

Lipschitz free boundaries are $C^1$ with Hölder continuous normal vectors.

Viscosity solutions satisfy the free boundary problem classically.

Regularity results hold in both spatial and temporal variables.

Abstract

This paper studies the regularity of the free boundary for viscosity solutions to a parabolic Bernoulli-type free boundary problem with variable coefficients. The main result is that Lipschitz free boundaries are $C^1$ with a normal vector that varies with a H\"older modulus of continuity in both space and in time. As a consequence, the viscosity solution satisfies the free boundary problem in a classical sense.