Nonlocal Harnack inequality in a disconnected region

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.

arXiv comment15 pages