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

  • Victor IvriiEmail author


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AM]
    W. Arendt; R. Mazzeo. Friedlander’s eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Commun. Pure Appl. Anal., 11(6):2201–2212 (2012).MathSciNetCrossRefGoogle Scholar
  2. [DG]
    J. J. Duistermaat; V. W. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math., 29(1):39–79 (1975).MathSciNetCrossRefGoogle Scholar
  3. [GP]
    A. Girouard; I. Polterovich. Spectral geometry of the Steklov problem arxiv:1411.6567
  4. [GLPS]
    A. Girouard; J. Lagacé; I. Polterovich, A. Savo. The Steklov spectrum of cuboids. arxiv:1711.03075
  5. [GG]
    G. Grubb. Mixed boundary problems on creased domains. Private Communication.Google Scholar
  6. [Ivr1]
    V. Ivrii. Microlocal Analysis, Sharp Spectral Asymptotics and Applications.Google Scholar
  7. [Ivr2]
    V. Ivrii. Spectral asymptotics for fractional Laplacians.Google Scholar
  8. [Ivr3]
    V. Ivrii. 100 years of Weyl’s law, Bulletin of Mathematical Sciences, Springer (2016).Google Scholar
  9. [Ivr4]
    V. Ivrii. Talk: Eigenvalue Asymptotics for Fractional Laplacians.Google Scholar
  10. [Ivr45]
    V. Ivrii. Talk: Eigenvalue asymptotics for Steklov’s problem in the domain with edges.Google Scholar
  11. [S-KP]
    M. Khalile; K. Pankrashkin. Eigenvalues of Robin Laplacians in infinite sectors. arxiv:1607.06848.
  12. [LPPS]
    M. Levitin, L. Parnovski; I. Polterovich; D. A. Sher. Sloshing, Steklov and corners I: Asymptotics of sloshing eigenvalues. arxiv:1709.01891
  13. [Nec]
    J. Necas. Direct methods in the theory of elliptic equations. Springer Monographs in Mathematics. Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner.CrossRefGoogle Scholar
  14. [PS]
    I. Polterovich; D. A. Sher. Heat invariants of the Steklov problem. J. Geom. Analysis 25 , no. 2, 924–950 (2015).MathSciNetCrossRefGoogle Scholar
  15. [Se]
    R. Seeley. A sharp asymptotic estimate for the eigenvalues of the Laplacian in a domain of  \(\varvec {R}^{3}\). Advances in Math., 102 (3):244–264 (1978).Google Scholar
  16. [SaVa]
    Yu. Safarov; D. Vassiliev. Branching Hamiltonian billiards. Soviet Math. Dokl. 38:64–68 (1989).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations