Boundary regularity for the 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 ${\mathbf{R}}^{n+1}$. 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.

40 pages

References in corpus (1)

• The tusk condition and Petrovski criterion for the normalized $p\mspace{1mu}$-parabolic equation

Cited by in corpus (4)

• Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior
• Existence of variational solutions to doubly nonlinear systems in nondecreasing domains
• Continuity up to the boundary for obstacle problems to porous medium type equations
• Existence of variational solutions to doubly nonlinear systems in general noncylindrical domains