Nonlocal Harnack inequality in a disconnected region

Se-Chan Lee; Se-Chan Lee

doi:10.3934/mine.2025028

Mathematics in Engineering

2025, Volume 7, Issue 6: 678-694. doi: 10.3934/mine.2025028

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Nonlocal Harnack inequality in a disconnected region

Se-Chan Lee ^,

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

Received: 05 September 2025 Revised: 03 November 2025 Accepted: 09 November 2025 Published: 11 November 2025

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.
- Harnack inequality,
- nonlocal equation,
- disconnected region,
- Poisson kernel,
- maximum principle
Citation: Se-Chan Lee. Nonlocal Harnack inequality in a disconnected region[J]. Mathematics in Engineering, 2025, 7(6): 678-694. doi: 10.3934/mine.2025028

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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.

References

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