Stable solutions of elliptic equations on Riemannian manifolds

Details Stable solutions of elliptic equations on Riemannian manifolds Stable solutions of elliptic equations on Riemannian manifolds

2008-09-17

arxiv.org/abs/0809.3025v1

Mathematics

Analysis of PDEs

Scientific paper

This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincar\'e inequality which allows to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.

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