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

  title={Friedlander ’ S Eigenvalue Inequalities and the Dirichlet-to-neumann Semigroup},
  author={Wolfgang Arendt and Rafe Mazzeo},
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… CONTINUE READING

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