Nonlocal Harnack inequality in a disconnected region

TL;DR

This paper proves a Harnack inequality for nonlocal equations in disconnected regions, revealing a purely nonlocal phenomenon that relates solutions across separate components, with two different proof methods.

Contribution

It introduces a novel Harnack inequality for nonlocal equations in disconnected regions, using maximum principle and Poisson kernel estimates.

Findings

Establishes a Harnack inequality for weak solutions in disconnected regions.

Demonstrates the inequality captures nonlocal effects with no local analogue.

Provides two different proof approaches for the inequality.

Abstract

We establish a Harnack inequality for weak solutions of nonlocal equations in a disconnected region. The inequality compares the value of a solution on one connected component with its value on another, capturing a purely nonlocal phenomenon with no local analogue. We provide two different approaches: one based on the localized maximum principle and another on the Poisson kernel estimates.