Weak Harnack estimates for a doubly nonlinear nonlocal p-Laplace equation

TL;DR

This paper proves new weak Harnack estimates with optimal tail behavior for weak supersolutions of a doubly nonlinear nonlocal p-Laplace equation, emphasizing local positivity and nonlocal features.

Contribution

It introduces a novel weak Harnack estimate framework for nonlocal doubly nonlinear equations, utilizing measure techniques and positivity expansion.

Findings

Establishment of weak Harnack estimates with optimal tail behavior

Highlighting the importance of local positivity over global positivity

Advancement in understanding nonlocal p-Laplace equations

Abstract

We establish a new type of weak Harnack estimates with optimal parabolic tail for the weak supersolutions to a doubly nonlinear nonlocal $p$-Laplace equation, which is modeled on the nonlocal Trudinger equation. Our results are achieved by employing the expansion of positivity and measure theoretical techniques. In particular, the weak Harnack estimates highlight the nonlocal feature, as we only require the local positivity of weak supersolutions instead of the global one.