Optimal Regularity for the parabolic No-Sign Obstacle Problem

TL;DR

This paper establishes optimal regularity results for the parabolic obstacle problem with a specific source term, and proves the free boundary is a $C^1$ graph near low energy points under Dini continuity assumptions.

Contribution

It proves optimal regularity for solutions and the $C^1$ regularity of the free boundary near low energy points for the parabolic obstacle problem.

Findings

Solution regularity matches the optimal bounds.

Free boundary is a $C^1$ graph near low energy points.

Results complete the theory for heat operator obstacle problems.

Abstract

We study the parabolic free boundary problem of obstacle type $$ \lap u-\frac{\partial u}{\partial t}= f\chi_{{u\ne 0}}. $$ Under the condition that $f=Hv$ for some function $v$ with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution $u$. Both the regularity and the assumptions are optimal. Using this result and assuming that $f$ is Dini continuous, we prove that the free boundary is, near so called low energy points, a $C^1$ graph. Our result completes the theory for this type of problems for the heat operator.