One-dimensional symmetry for integral systems in two dimensions

TL;DR

This paper proves one-dimensional symmetry results for stable solutions of nonlocal integral systems in two dimensions, extending De Giorgi type results without relying on local extension techniques.

Contribution

It establishes De Giorgi type symmetry results for nonlocal integral systems in 2D using direct integral estimates, without local extension methods.

Findings

Proves 1D symmetry for stable solutions in 2D nonlocal systems.

Extends De Giorgi conjecture results to nonlocal integral systems.

Develops new integral estimate techniques for nonlocal operators.

Abstract

The purpose of this brief paper is to prove De Giorgi type results for stable solutions of the following nonlocal system of integral equations in two dimensions $$ L(u_i) = H_i(u) \quad \text{in} \ \ \mathbb R^2 , $$ where $u=(u_i)_{i=1}^m$ for $u_i: \mathbb R^n\to \mathbb R$, $H=(H_i)_{i=1}^m$ is a general nonlinearity. The operator $L$ is given by $$L(u_i (x)):= \int_{\mathbb R^2} [u_i(x) - u_i(z)] K(z-x) dz,$$ for some kernel $K$. The idea is to apply a linear Liouville theorem for the quotient of partial derivatives, just like in the proof of the classical De Giorgi's conjecture in lower dimensions. Since there is no Caffarelli-Silvestre local extension problem associated to the above operator, we deal with this problem directly via certain integral estimates.