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

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

2011-05-13

arxiv.org/abs/1105.2711v2

Mathematics

Differential Geometry

26 pages

Scientific paper

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.

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