A comparison principle for the porous medium equation and its consequences

TL;DR

This paper establishes a comparison principle for the porous medium equation in general open sets, linking potential theory and variational inequalities, and explores properties of PME capacity including extremals for compact sets.

Contribution

It introduces a generalized comparison principle for the PME and connects potential theoretic and variational approaches to obstacle problems, advancing the theoretical framework.

Findings

Comparison principle valid in broader open sets

Connection between potential theory and variational inequalities

Existence of capacitary extremals for compact sets

Abstract

We prove a comparison principle for the porous medium equation in more general open sets in $\mathbb{R}^{n+1}$ than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.