Geodesic
Instability of elliptic equations on compact Riemannian manifolds with non-negative Ricci curvature
Electronic journal of differential equations, Tome 2010 (2010)
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.
Classification : 35J20, 58J05
Keywords: Riemannian manifold, Ricci curvature, local minimizer, gamma-convergence, reaction-diffusion equations 
@article{EJDE_2010__2010__a166,
     author = {Nascimento,  Arnaldo S. and Gon\c{c}alves,  Alexandre C.},
     title = {Instability of elliptic equations on compact {Riemannian} manifolds with non-negative {Ricci} curvature},
     journal = {Electronic journal of differential equations},
     year = {2010},
     volume = {2010},
     zbl = {1187.35036},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a166/}
}
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Nascimento,  Arnaldo S.; Gonçalves,  Alexandre C. Instability of elliptic equations on compact Riemannian manifolds with non-negative Ricci curvature. Electronic journal of differential equations, Tome 2010 (2010). http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a166/