Rigidity results for stable solutions of symmetric systems

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.