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

TL;DR

The paper tackles the weak Harnack problem for nonnegative weak supersolutions of the doubly nonlinear nonlocal $p$-Laplace equation $\partial_t(|u|^{p-2}u)+\mathcal{L}u=0$, where $\mathcal{L}$ is a fractional $p$-Laplacian-type operator with a kernel bounded by $|x-y|^{-(N+sp)}$. The authors develop a measure-theoretic framework, featuring a Caccioppoli-type energy estimate and a De Giorgi-type iteration, to prove two weak Harnack inequalities that incorporate the optimal parabolic tail and rely only on local positivity of the solution. A key innovation is the expansion of positivity, which propagates positivity in space-time while controlling nonlocal effects through the tail, enabling sharp, tail-sensitive lower bounds in time-space cylinders. These results advance the regularity theory for nonlocal, doubly nonlinear parabolic equations and reveal the precise role of time-dependent tails in nonlocal Harnack phenomena. The findings have implications for understanding diffusion processes with nonlocal interactions where positivity is established locally rather than globally.

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.

Theorems & Definitions (24)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Lemma 2.1
• Lemma 2.2: Lemma 2.2, BK24
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1