Boundary Regularity for the Porous Medium Equation
Björn, A., Björn, J., Gianazza, U., & Siljander, J. (2018). Boundary Regularity for the Porous Medium Equation. Archive for Rational Mechanics and Analysis, 230 (2), 493-538. doi:10.1007/s00205-018-1251-3
Published inArchive for Rational Mechanics and Analysis
© The Author(s) 2018
We study the boundary regularity of solutions to the porous medium equation ut=Δum 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 Rn+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. ...