Calculus of Variations and Geometric Measure Theory

A. Farina - L. Mari - E. Valdinoci

Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds

created by farina on 27 Jun 2013


Accepted Paper

Inserted: 27 jun 2013
Last Updated: 27 jun 2013

Journal: Comm. in PDEs
Year: 2013


Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation $-\Delta u = f(u)$ on a Riemannian manifold with non-negative Ricci curvature, we are able to classify both the solution and the manifold. We also discuss the classification of monotone (with respect to the direction of some Killing vector field) solutions, in the spirit of a conjecture of De Giorgi, and the rigidity features for overdetermined elliptic problems on submanifolds with boundary.