On The Weak Harnack Estimate For Nonlocal Equations

We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation \begin{align*} \partial_t(|u|^{p-2}u) + (-\Delta_p)^s u = 0 \end{align*} for \(p\in (1,\infty)\) and \(s \in (0,1)\). Our proof relies on expansion of positivity arguments...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2023-06
Main Author: Prasad, Harsh
Format: Article
Language:English
Subjects:
Estimates
Iterative methods
Laplace equation
Thermodynamics
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation \begin{align*} \partial_t(|u|^{p-2}u) + (-\Delta_p)^s u = 0 \end{align*} for \(p\in (1,\infty)\) and \(s \in (0,1)\). Our proof relies on expansion of positivity arguments developed by DiBenedetto, Gianazza and Vespri adapted to the nonlocal setup. Even in the linear case of the nonlocal heat equation and in the time-independent case of fractional \(p-\)Laplace equation, our approach provides an alternate route to Harnack estimates without using Moser iteration, log estimates or Krylov-Safanov covering arguments.
ISSN:2331-8422