Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds

Alberto Farina, Luciano Mari, Enrico Valdinoci

Research output: Contribution to journalArticlepeer-review

35 Citations (Scopus)


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 - Δu = 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.

Original languageEnglish
Pages (from-to)1818-1862
Number of pages45
JournalCommunications in Partial Differential Equations
Issue number10
Publication statusPublished - Oct 2013
Externally publishedYes


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