Harmonic approximation and improvement of flatness in a singular perturbation problem

Abstract

We study the De Giorgi type conjecture, that is, one dimensional symmetry problem for entire solutions of an two components elliptic system in $\mathbb{R}^n$, for all $n\geq 2$. We prove that, if a solution $(u,v)$ has a linear growth at infinity, then it is one dimensional, that is, depending only on one variable. The main ingredient is an improvement of flatness estimate, which is achieved by the harmonic approximation technique adapted in the singularly perturbed situation.