Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup (original) (raw)

If Ω is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary Γ = ∂Ω, the Dirichlet-to-Neumann operator D λ is defined on L 2 (Γ) for any real λ. We prove a close relationship between the eigenvalues of D λ and those of the Robin Laplacian ∆µ, i.e. the Laplacian with Robin boundary conditions ∂ν u = µu. This is used to give another proof of the Friedlander inequalities between Neumann and Dirichlet eigenvalues, λ N k+1 ≤ λ D k , k ∈ N, and to sharpen the inequality to be strict, whenever Ω is a Lipschitz domain in R d. We give new counterexamples to these inequalities in the general Riemannian setting. Finally, we prove that the semigroup generated by −D λ , for λ sufficiently small or negative, is irreducible.