Instability results for an elliptic equation on compact Riemannian manifolds with non-negative Ricci curvature

TL;DR

This paper investigates the stability of solutions to an elliptic equation on compact Riemannian manifolds, showing nonexistence of nonconstant local minimizers under non-negative Ricci curvature, and existence under certain negative curvature conditions.

Contribution

It establishes conditions for the existence and nonexistence of nonconstant local minimizers of a class of functionals on Riemannian manifolds, linking curvature properties to solution stability.

Findings

Nonexistence of nonconstant local minimizers on manifolds with non-negative Ricci curvature.

Existence of nonconstant local minimizers on surfaces with negative Gauss curvature along a simple closed geodesic.

Results connect geometric curvature conditions with variational stability properties.

Abstract

We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative we prove existence of nonconstant local minimizers for the same class of functionals.