New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary

Jimmy Lamboley (1) , Pieralberto Sicbaldi (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 I2M - Institut de Mathématiques de Marseille

We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifolds with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.

Domaines: Optimisation et contrôle [math.OC]; Géométrie différentielle [math.DG]

Jimmy Lamboley, Pieralberto Sicbaldi. New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary. International Mathematics Research Notices, 2014, 2015, pp.8752-8798. ⟨10.1093/imrn/rnu211⟩. ⟨hal-01074375⟩