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Lower semicontinuity of weak supersolutions to the porous medium equation


Authors: Benny Avelin and Teemu Lukkari
Journal: Proc. Amer. Math. Soc. 143 (2015), 3475-3486
MSC (2010): Primary 35K55, 31C45
DOI: https://doi.org/10.1090/proc/12727
Published electronically: April 23, 2015
MathSciNet review: 3348790
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Abstract: Weak supersolutions to the porous medium equation are defined by means of smooth test functions under an integral sign. We show that nonnegative weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. This shows that weak supersolutions belong to a class of supersolutions defined by a comparison principle.


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Additional Information

Benny Avelin
Affiliation: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
Email: benny.avelin@math.uu.se

Teemu Lukkari
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland
Email: teemu.j.lukkari@jyu.fi

DOI: https://doi.org/10.1090/proc/12727
Keywords: Porous medium equation, supersolutions, comparison principle, lower semicontinuity, degenerate diffusion
Received by editor(s): December 10, 2013
Published electronically: April 23, 2015
Additional Notes: The research reported in this work was done during the authors’ stay at the Institut Mittag-Leffler (Djursholm, Sweden).
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society