Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature

TL;DR

This paper develops a geometric Poincaré-type formula and classifies stable solutions to linear elliptic boundary value problems with nonlinear Robin conditions on Riemannian manifolds with nonnegative Ricci curvature, refining previous results.

Contribution

It introduces a new geometric formula and provides a refined classification of stable solutions for elliptic problems with nonlinear boundary conditions on specific Riemannian manifolds.

Findings

Established a geometric Poincaré-type formula.

Classified stable solutions under nonnegative Ricci curvature.

Refined previous classification results.

Abstract

We present a geometric formula of Poincar\'e type, which is inspired by a classical work of Sternberg and Zumbrun, and we provide a classification result of stable solutions of linear elliptic problems with nonlinear Robin conditions on Riemannian manifolds with nonnegative Ricci curvature. The result obtained here is a refinement of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.