A one-dimensional symmetry result for solutions of an integral equation of convolution type

TL;DR

This paper proves that solutions to a specific convolution-type integral equation in the plane, which are monotone or stable, must depend on only one spatial variable, revealing a one-dimensional symmetry property.

Contribution

The paper establishes a new one-dimensional symmetry result for monotone or stable solutions of a convolution-type integral equation in the plane.

Findings

Monotone solutions are necessarily one-dimensional.

Stable solutions are necessarily one-dimensional.

The result applies to a class of integral equations of convolution type.

Abstract

We consider an integral equation in the plane, in which the leading operator is of convolution type, and we prove that monotone (or stable) solutions are necessarily one-dimensional.