Stable solutions of symmetric systems on Riemannian manifolds

Abstract

Abstract. We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold M without boundary,when ∆g stands for the Laplace-Beltrami operator,This system is called symmetric if the matrix of partial derivatives of all components of H, that is H(u) = (∂ j H i (u)) m i,j=1 , 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.

Rigidity results in diffusion Markov triples

“…The advantage of this method is that the weights are nonnegative and possess a geometric interpretation, hence the possible vanishing of the integral in the Poincaré-type inequalities implies the vanishing of the corresponding geometric weight, which in turn provides a series of useful geometric rigidities. Rigidity results via Poincaré-type inequalities have been recently obtained in different settings, including, among the others, systems of equations [14,27,28], manifolds [23,24,21,4,26,18], stratified groups [30,42,6,29], equations with drift [22], stratified media [44,13,16] and fractional equations [45,46,15], and there are also applications for equations in infinite dimensional spaces [10]. The method can be also applied to deduce new weighted Poincaré inequalities from the explicit knowledge of a stable solution, see [31], and it is also flexible enough to deal with Neumann boundary conditions [3,17].…”

Stable solutions of symmetric systems involving hypoelliptic operators

“…However, the reverse is not necessarily true. We now provide the notion of symmetric systems introduced in [25,27] when Ω = R n . Symmetric systems play a fundamental role throughout this paper when we study system (1.10) with a general nonlinearity H(u) = (H i (u)) m i=1 .…”

“…In addition, in a series of articles in [3][4][5] symmetry results and Liouville theorems, amongst other results, are proved. The author considered the above system (1.28) on Riemannian manifolds in [25] and established a geometrical inequality to study the level set of solutions and to prove Liouville theorems.…”