Friedlander’s Eigenvalue Inequalities and the Dirichlet-to-neumann Semigroup

If Ω is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary Γ = ∂Ω, the Dirichlet-to-Neumann operator Dλ is defined on L2(Γ) for any real λ. We prove a close relationship between the eigenvalues of Dλ and those of the Robin Laplacian ∆μ, i.e. the Laplacian with Robin boundary conditions ∂νu = μu. This is used to give another proof of the Friedlander inequalities between Neumann and Dirichlet eigenvalues, λk+1 ≤ λk , k ∈ N, and to sharpen the inequality to be strict, whenever Ω is a Lipschitz domain in Rd. We give new counterexamples to these inequalities in the general Riemannian setting. Finally, we prove that the semigroup generated by −Dλ, for λ sufficiently small or negative, is irreducible.