Boundary Regularity for the Porous Medium Equation
Anders Björn, Jana Björn, Ugo Gianazza, Juhana Siljander · Archive for Rational Mechanics and Analysis · 2018
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/superparabolic functions and weak sub/supersolutions.