On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

Abstract

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.

26 pages

References in corpus (2)

• A Reilly formula and eigenvalue estimates for differential forms
• The complete Dirichlet-to-Neumann map for differential forms

Cited by in corpus (6)

• Higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues
• Escobar's Conjecture on a sharp lower bound for the first nonzero Steklov eigenvalue
• Estimates for higher Steklov eigenvalues
• A higher dimensional generalization of Hersch-Payne-Schiffer inequality for Steklov eigenvalues
• Robin and Steklov isospectral manifolds
• On the spectrum of the Dirichlet-to-Neumann operator acting on forms of a Euclidean domain