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HOME > Table of Contents and Abstracts > Vol. 65, No. 4
Tohoku Mathematical Journal
2013
December
SECOND SERIES VOL. 65, NO. 4
Tohoku Math. J.
65 (2013), 569-589
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Title
ON THE HARNACK INEQUALITY FOR PARABOLIC MINIMIZERS IN METRIC MEASURE SPACES
Author
Niko Marola and Mathias Masson
(Received August 16, 2012, revised March 7, 2013) |
Abstract.
In this note we consider problems related to parabolic partial differential equations in geodesic metric measure spaces, that are equipped with a doubling measure and a Poincaré inequality. We prove a location and scale invariant Harnack inequality for a minimizer of a variational problem related to a doubly non-linear parabolic equation involving the $p$-Laplacian. Moreover, we prove the sufficiency of the Grigor'yan--Saloff-Coste theorem for general $p>1$ in geodesic metric spaces. The approach used is strictly variational, and hence we are able to carry out the argument in the metric setting.
Mathematics Subject Classification.
Primary: 35B65, Secondary: 35K55, 49N60.
Key words and phrases.
Doubling measure, Harnack inequality, metric space, minimizer, Newtonian space, parabolic, Poincaré inequality.
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