Symmetry of odd solutions to equations with fractional Laplacian

TL;DR

This paper proves symmetry and monotonicity properties of odd solutions to equations involving the fractional Laplacian in domains with perpendicular symmetries, expanding understanding of solution behavior under these conditions.

Contribution

It establishes new symmetry and monotonicity results for solutions with fractional Laplacian, including conditions for symmetry in perpendicular directions.

Findings

Solutions are symmetric in the perpendicular direction under given conditions.

Solutions are monotonic where they have a fixed sign.

Applicable to a class of examples with specific symmetry properties.

Abstract

We present a symmetry result to solutions of equations involving the fractional Laplacian in a domain with at least two perpendicular symmetries. We show that if the solution is continuous, bounded, and odd in one direction such that it has a fixed sign on one side, then it will be symmetric in the perpendicular direction. Moreover, the solution will be monotonic in the part where it is of fixed sign. In addition, we present also a class of examples in which our result can be applied.