Boundary regularity for the porous medium equation

TL;DR

This paper investigates the boundary regularity of solutions to the porous medium equation for m>1, establishing optimal conditions for boundary value attainment and developing a new comparison principle.

Contribution

It introduces a barrier characterization of boundary regularity for general domains and a new strict comparison principle for sub- and superparabolic functions.

Findings

Boundary regularity characterized by elliptic Wiener criterion.

Established a new strict comparison principle.

Developed comprehensive boundary regularity theory for porous medium equation.

Abstract

We study the boundary regularity of solutions to the porous medium equation $u_t = \Delta u^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the parabolic boundary has a solution which attains the boundary values, provided that the spatial domain satisfies the elliptic Wiener criterion. This condition is known to be optimal, and it is a consequence of our main theorem which establishes a barrier characterization of regular boundary points for general -- not necessarily cylindrical -- domains in ${\bf R}^{n+1}$. One of our fundamental tools is a new strict comparison principle between sub- and superparabolic functions, which makes it essential for us to study both nonstrict and strict Perron solutions to be able to develop a fruitful boundary regularity theory. Several other comparison principles and pasting lemmas are also obtained. In the process we obtain a rather complete picture of the relation between sub/super\-para\-bolic functions and weak sub/super\-solu\-tions.