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Symmetry of solutions to nonlocal Neumann problems with the fractional Laplacian in half spaces
Abstract
This paper investigates the fractional Laplacian equation in half-spaces, focusing on cases with nonlocal Neumann boundary conditions. We consider both single equations and systems, and establish two key principles: the decay at infinity principle and the narrow region principle. These foundational results, combined with the direct method of moving planes, allow us to derive symmetry and monotonicity properties for decaying solutions. To the best of our knowledge, these are the first results concerning the symmetry of solutions to fractional Laplacian equations with nonlocal Neumann conditions.
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BibTeX
@misc{liu2026symmetry,
title = {{Symmetry of solutions to nonlocal Neumann problems with the fractional Laplacian in half spaces}},
author = {Baiyu Liu and Wenli Tan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33103},
eprint = {2609.33103},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33103},
}
RIS
TY - UNPB
TI - Symmetry of solutions to nonlocal Neumann problems with the fractional Laplacian in half spaces
AU - Baiyu Liu
AU - Wenli Tan
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33103
UR - https://arxiv.org/abs/2609.33103
AB - This paper investigates the fractional Laplacian equation in half-spaces, focusing on cases with nonlocal Neumann boundary conditions. We consider both single equations and systems, and establish two key principles: the decay at infinity principle and the narrow region principle. These foundational results, combined with the direct method of moving planes, allow us to derive symmetry and monotonicity properties for decaying solutions. To the best of our knowledge, these are the first results concerning the symmetry of solutions to fractional Laplacian equations with nonlocal Neumann conditions.
ER -