Boundary Regularity for the Porous Medium Equation

Abstract

We study the boundary regularity of solutions to the porous medium equation ut = ∆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 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/superparabolic functions and weak sub/supersolutions.

On the Hölder regularity for obstacle problems to porous medium type equations

“…For the porous medium equation, a crucial approximation scheme in this connection exploits the obstacle problem in [19]. As further applications, we mention questions related to boundary regularity addressed in [3,21]. In this context, we also point out that there is an alternative approach to the obstacle problem, in which the solution is defined as the smallest weak supersolution lying above the given obstacle ψ.…”