The equivalence of weak and very weak supersolutions to the porous medium equation

TL;DR

This paper demonstrates the equivalence of different supersolution concepts for the porous medium equation, using comparison principles and a Schwarz type method, and establishes boundary regularity of Perron solutions.

Contribution

It establishes the equivalence of weak, very weak, and $m$-superporous supersolutions for the porous medium equation under suitable conditions.

Findings

Various notions of supersolutions are equivalent.

Comparison principles and Schwarz type methods are effective tools.

Perron solutions with continuous boundary data are continuous up to the boundary.

Abstract

We prove that various notions of supersolutions to the porous medium equation are equivalent under suitable conditions. More spesifically, we consider weak supersolutions, very weak supersolutions, and $m$-superporous functions defined via a comparison principle. The proofs are based on comparison principles and a Schwarz type alternating method, which are also interesting in their own right. Along the way, we show that Perron solutions with merely continuous boundary values are continuous up to the parabolic boundary of a sufficiently smooth space-time cylinder.