Revista Matemática Iberoamericana


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Volume 33, Issue 2, 2017, pp. 573–594
DOI: 10.4171/RMI/950

A comparison principle for the porous medium equation and its consequences

Benny Avelin[1] and Teemu Lukkari[2]

(1) Department of Mathematics, Uppsala Universitet, P.O. Box 480, 75106, UPPSALA, SWEDEN
(2) Department of Mathematics, Aalto University, P.O. Box 11100, 00076, AALTO UNIVERSITY, FINLAND

We prove a comparison principle for the porous medium equation in more general open sets in $\mathbb R^{n+1}$ than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.

Keywords: Porous medium equation, PME, elliptic comparison principle, obstacle problem, parabolic capacity

Avelin Benny, Lukkari Teemu: A comparison principle for the porous medium equation and its consequences. Rev. Mat. Iberoamericana 33 (2017), 573-594. doi: 10.4171/RMI/950