Regularity of Solutions to a Parabolic Free Boundary Problem with Variable Coefficients

TL;DR

This paper proves that viscosity solutions to a parabolic free boundary problem with variable coefficients are Lipschitz continuous, establishing the optimal regularity under certain conditions.

Contribution

It demonstrates the optimal Lipschitz regularity of solutions to a parabolic free boundary problem with variable coefficients, given a Lipschitz free boundary and non-degeneracy.

Findings

Viscosity solutions are Lipschitz continuous.

Regularity is optimal under given assumptions.

Results apply to variable coefficient problems.

Abstract

The main result of this paper is to prove that viscosity solutions to a parabolic free boundary problem with variable coefficients are Lipschitz continuous under the assumptions that the solution has a Lipschitz free boundary and satisfies a non-degeneracy condition. This is the optimal regularity for this problem.