Harmonic approximation and improvement of flatness in a singular perturbation problem

TL;DR

This paper proves that solutions with linear growth to a two-component elliptic system are one-dimensional, using harmonic approximation techniques to improve flatness estimates in a singular perturbation context.

Contribution

It introduces an improved flatness estimate via harmonic approximation for singular perturbation problems, establishing one-dimensional symmetry under linear growth conditions.

Findings

Solutions with linear growth are one-dimensional.

Harmonic approximation effectively improves flatness estimates.

The approach applies to singularly perturbed elliptic systems.

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.