Lower semicontinuous obstacles for the porous medium equation

TL;DR

This paper studies the obstacle problem for the porous medium equation with irregular, lower semicontinuous obstacles, establishing existence of solutions and approximation results in the slow diffusion regime.

Contribution

It introduces a novel approach to handle irregular obstacles in the porous medium equation, proving existence of minimal supersolutions and approximation by bounded supersolutions.

Findings

Existence of solutions as minimal supersolutions above irregular obstacles

Approximation of non-negative supersolutions by bounded supersolutions

Handling of obstacles with only boundedness and lower semicontinuity

Abstract

We deal with the obstacle problem for the porous medium equation in the slow diffusion regime $m>1$. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.