Instability of elliptic equations on compact Riemannian manifolds with non-negative Ricci curvature

Abstract

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

Keywords

• Riemannian manifold
• Ricci curvature
• local minimizer
• Gamma-convergence
• reaction-diffusion equations