A weak Harnack inequality for fractional evolution equations with discontinuous coefficients
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 4, p. 903-940

We study linear time fractional diffusion equations in divergence form of time order less than one. It is merely assumed that the coefficients are measurable and bounded, and that they satisfy a uniform parabolicity condition. As a main result we establish for nonnegative weak supersolutions of such problems a weak Harnack inequality with optimal critical exponent. The proof relies on new a priori estimates for time fractional problems and uses Moser’s iteration technique and an abstract lemma of Bombieri and Giusti, the latter allowing to avoid the rather technically involved approach via $BMO$. As applications of the weak Harnack inequality we establish the strong maximum principle, the continuity of weak solutions at $t=0$, and a uniqueness theorem for global bounded weak solutions.

Published online : 2019-02-21
Classification:  35R09,  45K05
@article{ASNSP_2013_5_12_4_903_0,
author = {Zacher, Rico},
title = {A weak Harnack inequality for fractional evolution equations with discontinuous coefficients},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 12},
number = {4},
year = {2013},
pages = {903-940},
zbl = {1285.35124},
mrnumber = {3184573},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2013_5_12_4_903_0}
}

Zacher, Rico. A weak Harnack inequality for fractional evolution equations with discontinuous coefficients. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 4, pp. 903-940. http://www.numdam.org/item/ASNSP_2013_5_12_4_903_0/

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