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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 \({{\bf R}^{n+1}}\). 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.

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Correspondence to Jana Björn.

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Communicated by P. -L. Lions

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Björn, A., Björn, J., Gianazza, U. et al. Boundary Regularity for the Porous Medium Equation. Arch Rational Mech Anal 230, 493–538 (2018). https://doi.org/10.1007/s00205-018-1251-3

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  • DOI: https://doi.org/10.1007/s00205-018-1251-3