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Proceedings of the American Mathematical Society

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A boundary Harnack inequality for singular equations of $ p$-parabolic type


Authors: Tuomo Kuusi, Giuseppe Mingione and Kaj Nyström
Journal: Proc. Amer. Math. Soc. 142 (2014), 2705-2719
MSC (2010): Primary 35K10, 35K67, 35K92, 35B65
DOI: https://doi.org/10.1090/S0002-9939-2014-12171-X
Published electronically: April 29, 2014
MathSciNet review: 3209326
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Abstract: We prove a boundary Harnack type inequality for nonnegative solutions to singular equations of $ p$-parabolic type, $ 2n/(n+1)<p<2$, in a time-independent cylinder whose base is $ C^{1,1}$-regular. Simple examples show, using the corresponding estimates valid for the heat equation as a point of reference, that this type of inequality cannot, in general, be expected to hold in the degenerate case ( $ 2<p<\infty $).


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Additional Information

Tuomo Kuusi
Affiliation: Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland
Email: tuomo.kuusi@aalto.fi

Giuseppe Mingione
Affiliation: Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/a, Campus, 43124 Parma, Italy
Email: giuseppe.mingione@unipr.it

Kaj Nyström
Affiliation: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
Email: kaj.nystrom@math.uu.se

DOI: https://doi.org/10.1090/S0002-9939-2014-12171-X
Keywords: $p$-parabolic equation, degenerate, singular, Harnack inequality, boundary Harnack inequality, Lipschitz domain $C^{1,\alpha}$-domain, $C^{1, 1}$
Received by editor(s): August 29, 2012
Published electronically: April 29, 2014
Communicated by: Tatiana Toro
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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