One-dimensional symmetry for integral systems in two dimensions

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.