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THE STEKLOV SPECTRUM OF CUBOIDS

Part of: Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  06 December 2018

Alexandre Girouard ,
Jean Lagacé ,
Iosif Polterovich  and
Alessandro Savo
Alexandre Girouard
Affiliation:
Département de mathématiques et de statistique, Pavillon Alexeandre-Vachon, Université Laval, Québec, QC G1V 0A6, Canada email alexandre.girouard@mat.ulaval.ca
Jean Lagacé
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC H3C 3J7, Canada
Iosif Polterovich
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC H3C 3J7, Canada email iossif@dms.umontreal.ca
Alessandro Savo
Affiliation:
Dipartimento SBAI, Sezione di Matematica Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Roma, Italy email alessandro.savo@sbai.uniroma1.it
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Abstract

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension $d\geqslant 3$. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the $(d-2)$-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids.

MSC classification

Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Type
Research Article
Information
Mathematika , Volume 65 , Issue 2 , 2019 , pp. 272 - 310
Copyright
Copyright © University College London 2018 

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Footnotes

1

Current address: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K. email j.lagace@ucl.ac.uk

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