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Spectral Asymptotics for Dirichlet to Neumann Operator in the Domains with Edges

  • Victor IvriiEmail author
Chapter

Abstract

We consider eigenvalues of the Dirichlet-to-Neumann operator for Laplacian in the domain (or manifold) with edges and establish the asymptotics of the eigenvalue counting function
$$N(\lambda )= \kappa _0\lambda ^d +O(\lambda ^{d-1})\qquad \text {as} \lambda \rightarrow +\infty $$
where d is dimension of the boundary. Further, in certain cases we establish two-term asymptotics
$$N(\lambda )=\kappa _0\lambda ^d+\kappa _1\lambda ^{d-1}+o(\lambda ^{d-1})\qquad \text {as} \lambda \rightarrow +\infty $$
We also establish improved asymptotics for Riesz means.

Key words and phrases

Dirichlet-to-Neumann operator spectral asymptotics 

2010 Mathematics Subject Classification:

35P20 58J50 

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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