De Giorgi type results for elliptic systems

TL;DR

This paper extends De Giorgi type results to elliptic systems, showing that solutions with monotone components are one-dimensional under certain conditions, using Liouville theorems and geometric inequalities.

Contribution

It introduces the concept of orientable systems and extends De Giorgi results to systems with multiple equations, including stable solutions and rigidity properties.

Findings

Solutions with monotone components are one-dimensional in low dimensions.

Extension of geometric Poincaré inequality to systems.

Stable solutions exhibit parallel gradients among components.

Abstract

We consider the following elliptic system \Delta u =\nabla H (u) \ \ \text{in}\ \ \mathbf{R}^N, where $u:\mathbf{R}^N\to \mathbf{R}^m$ and $H\in C^2(\mathbf{R}^m)$, and prove, under various conditions on the nonlinearity $H$ that, at least in low dimensions, a solution $u=(u_i)_{i=1}^m$ is necessarily one-dimensional whenever each one of its components $u_i$ is monotone in one direction. Just like in the proofs of the classical De Giorgi's conjecture in dimension 2 (Ghoussoub-Gui) and in dimension 3 (Ambrosio-Cabr\'{e}), the key step is a Liouville theorem for linear systems. We also give an extension of a geometric Poincar\'{e} inequality to systems and use it to establish De Giorgi type results for stable solutions as well as additional rigidity properties stating that the gradients of the various components of the solutions must be parallel. We introduce and exploit the concept of {\it an orientable system}, which seems to be key for dealing with systems of three or more equations. For such systems, the notion of a stable solution in a variational sense coincide with the pointwise (or spectral) concept of stability.