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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.

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