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Archive for Rational Mechanics and Analysis

, Volume 230, Issue 2, pp 493–538 | Cite as

Boundary Regularity for the Porous Medium Equation

  • Anders Björn
  • Jana Björn
  • Ugo Gianazza
  • Juhana Siljander
Open Access
Article
  • 210 Downloads

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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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsLinköping UniversityLinköpingSweden
  2. 2.Department of Mathematics “F. Casorati”Università di PaviaPaviaItaly
  3. 3.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland

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