On the De Giorgi type conjecture for an elliptic system modeling phase separation

TL;DR

This paper proves that positive solutions to a specific elliptic system modeling phase separation are one-dimensional if they are local minimizers with linear growth, establishing symmetry and Lipschitz regularity.

Contribution

It demonstrates one-dimensional symmetry for local minimizers with linear growth in a phase separation elliptic system, extending De Giorgi type conjectures.

Findings

Solutions are one-dimensional if they are local minimizers with linear growth.

Global Lipschitz continuity of solutions is established under linear growth.

The result applies to solutions in all dimensions n ≥ 2.

Abstract

In this paper we study the one dimensional symmetry problem of entire solutions to the problem \[\Delta u=uv^2,\Delta v=vu^2,u,v>0 \text{in} \mathbb{R}^n,\] for all $n\geq 2$. We prove that, if a solution $(u,v)$ is a local minimizer and has linear growth at infinity, then it is one dimensional, i.e. depending only on one variable. In the proof we also obtain the global Lipschitz continuity of solutions only under the linear growth assumption.