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.