Stable solutions of symmetric systems on Riemannian manifolds

Abstract:

We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold $\mathbb {M}$ without boundary, \begin{equation*} -\Delta _g u_i = H_i(u_1,\cdots ,u_m) \ \ \text {on} \ \ \mathbb {M}, \end{equation*} when $\Delta _g$ stands for the Laplace-Beltrami operator, $u_i:\mathbb {M}\to \mathbb R$ and $H_i\in C^1(\mathbb R^m)$ for $1\le i\le m$. This system is called symmetric if the matrix of partial derivatives of all components of $H$, that is, $\mathbb H(u)=(\partial _j H_i(u))_{i,j=1}^m$, is symmetric. We prove a stability inequality and a Poincaré type inequality for stable solutions using the Bochner-Weitzenböck formula. Then, we apply these inequalities to establish Liouville theorems and flatness of level sets for stable solutions of the above symmetric system under certain assumptions on the manifold and on solutions.