A weak Harnack inequality for fractional evolution equations with discontinuous coefficients

TL;DR

This paper establishes a weak Harnack inequality for nonnegative solutions of linear time fractional diffusion equations with discontinuous coefficients, enabling further results like maximum principles and solution regularity.

Contribution

It introduces a novel weak Harnack inequality for fractional diffusion equations with minimal coefficient regularity, using new a priori estimates and avoiding complex BMO techniques.

Findings

Proved a weak Harnack inequality with optimal critical exponent.

Established the strong maximum principle for fractional diffusion.

Demonstrated continuity of solutions at initial time t=0.

Abstract

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 the 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, continuity of weak solutions at t = 0, and a Liouville type theorem.