Isoperimetric control of the Steklov spectrum

Bruno Colbois (1) , Ahmad El Soufi (2) , Alexandre Girouard (1)

1 Laboratoire de Mathématiques
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus. We also establish a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.

Mots clés: Laplacian; eigenvalue; submanifold; isoperimetric ratio

Domaines: Géométrie métrique [math.MG]

Bruno Colbois, Ahmad El Soufi, Alexandre Girouard. Isoperimetric control of the Steklov spectrum. Journal of Functional Analysis, 2011, 261 (5), pp.1384-1399. ⟨hal-00779282⟩