# Optimal regularity for the parabolic no-sign obstacle type problem

### John Andersson

KTH Royal Institute of Technology, Stockholm, Sweden### Erik Lindgren

KTH Royal Institute of Technology, Stockholm, Sweden### Henrik Shahgholian

KTH Royal Institute of Technology, Stockholm, Sweden

## Abstract

We study the parabolic free boundary problem of obstacle type

$\Delta u-\frac{\partial u}{\partial t}= f\chi_{\{u\ne 0\}}.$

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.

## Cite this article

John Andersson, Erik Lindgren, Henrik Shahgholian, Optimal regularity for the parabolic no-sign obstacle type problem. Interfaces Free Bound. 15 (2013), no. 4, pp. 477–499

DOI 10.4171/IFB/311