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. © 2013 Taylor and Francis Group, LLC.

Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds / A. Farina, L. Mari, E. Valdinoci. - In: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0360-5302. - 38:10(2013), pp. 1818-1862. [10.1080/03605302.2013.795969]

Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds

L. Mari
Secondo
;
E. Valdinoci
Ultimo
2013

Abstract

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. © 2013 Taylor and Francis Group, LLC.
Partial differential equations on manifolds; Rigidity and classification results
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/255035
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