Rigidity results for stable solutions of symmetric systems

TL;DR

This paper introduces the concept of symmetric systems for nonlinear elliptic systems and proves Liouville and regularity theorems for stable solutions, advancing understanding of solution behavior in various domains.

Contribution

It defines symmetric systems and establishes Liouville and regularity results for stable solutions, extending previous linear Liouville theorems to nonlinear systems.

Findings

Symmetric systems are crucial for Liouville and regularity theorems.

Improved linear Liouville theorem for nonlinear systems.

Stable solutions exhibit regularity under symmetry conditions.

Abstract

We study stable solutions of the following nonlinear system $$ -\Delta u = H(u) \quad \text{in} \ \ \Omega$$ where $u:\mathbb R^n\to \mathbb R^m$, $H:\mathbb R^m\to \mathbb R^m$ and $\Omega$ is a domain in $\mathbb R^n$. We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of $H$ is symmetric. It seems that this concept is crucial to prove Liouville theorems, when $\Omega=\mathbb R^n$, and regularity results, when $\Omega=B_1$, for stable solutions of the above system for a general nonlinearity $H \in C^1(\mathbb R ^m)$. Moreover, we provide an improvement for a linear Liouville theorem given in [20] that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.