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


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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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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