Rigidity Results for Elliptic Boundary Value Problems: Stable Solutions for Quasilinear Equations with Neumann or Robin Boundary Conditions

摘要

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