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 Poincaré 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. From 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.

Rigidity results for elliptic boundary value problems: Stable solutions for quasilinear equations with Neumann or Robin boundary conditions / Pinamonti, Andrea; Dipierro, Serena; Valdinoci, Enrico. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1687-0247. - 2020:5(2020), pp. 1366-1384. [10.1093/imrn/rny055]

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

Pinamonti, Andrea;Valdinoci, Enrico
2020-01-01

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 Poincaré 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. From 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.
2020
5
Pinamonti, Andrea; Dipierro, Serena; Valdinoci, Enrico
Rigidity results for elliptic boundary value problems: Stable solutions for quasilinear equations with Neumann or Robin boundary conditions / Pinamonti, Andrea; Dipierro, Serena; Valdinoci, Enrico. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1687-0247. - 2020:5(2020), pp. 1366-1384. [10.1093/imrn/rny055]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/209879
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