A weak Harnack inequality for fractional evolution equations with discontinuous coefficients

Zacher, Rico

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 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.

Classification : 35R09, 45K05

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