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Research Article | Open Access

Nonlocal Harnack inequality in a disconnected region

School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Korea
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Abstract

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

Keywords

Harnack inequalitynonlocal equationdisconnected regionPoisson kernelmaximum principle

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Mathematics in Engineering
Volume 7 Issue 6,
December 2025
Pages 678-694
DOI: 10.3934/mine.2025028

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Cite this article:
Lee S-C. Nonlocal Harnack inequality in a disconnected region. Mathematics in Engineering, 2025, 7(6): 678-694. https://doi.org/10.3934/mine.2025028

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Received: 05 September 2025
Revised: 03 November 2025
Accepted: 09 November 2025
Published: 15 December 2025
©2025 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)