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arXiv · 1709.07934

Rigidity results for elliptic boundary value problems: stable solutions for quasilinear equations with Neumann or Robin boundary conditions

Serena Dipierro·Andrea Pinamonti·Enrico Valdinoci
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Abstract

We provide a general approach to the classification results of stable solutions of (possibly nonlinear) elliptic problems with Robin conditions. The method is based on a geometric formula of Poincar\'e type, which is inspired by a classical work of Sternberg and Zumbrun and which gives an accurate description of the curvatures of the level sets of the stable solutions. {F}rom this, we show that the stable solutions of a quasilinear problem with Neumann data are necessarily constant. As a byproduct of this, we obtain an alternative proof of a celebrated result of Casten and Holland, and Matano. In addition, we will obtain as a consequence a new proof of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.

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Serena Dipierro, Andrea Pinamonti, Enrico Valdinoci. 2017-09-22. Rigidity results for elliptic boundary value problems: stable solutions for quasilinear equations with Neumann or Robin boundary conditions. https://arxiv.org/abs/1709.07934

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