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

TL;DR

This paper introduces a geometric approach to classify stable solutions of elliptic boundary value problems with Robin conditions, proving that solutions with Neumann data are constant and providing new proofs for existing results.

Contribution

It presents a novel geometric formula-based method for classifying stable solutions of elliptic problems with Robin conditions, including a new proof of known results.

Findings

Stable solutions with Neumann data are necessarily constant.

Provides an alternative proof of a classical result by Casten and Holland, and Matano.

Offers a new proof of recent results by Bandle et al.

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.