Extremal domains for the first eigenvalue in a general compact Riemannian manifold

Erwann Delay (1) , Pieralberto Sicbaldi (2)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
2 I2M - Institut de Mathématiques de Marseille

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the Riemannian manifold was required.

Domaines: Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]

Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete and Continuous Dynamical Systems - Series A, 2015, 35 (12), pp.5799-5825. ⟨10.3934/dcds.2015.35.5799⟩. ⟨hal-01266512⟩