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The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript

Girouard, Alexandre and Karpukhin, Mikhail and Levitin, Michael and Polterovich, Iosif (2022) The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript. Journal of Spectral Theory, 12 (1). pp. 195-225. ISSN 1664-039X. doi:10.4171/jst/399.

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How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander’s approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Girouard, Alexandre0000-0001-8823-831X
Karpukhin, Mikhail0000-0003-1935-731X
Levitin, Michael0000-0003-0020-3265
Additional Information:© 2022 European Mathematical Society. Published by EMS Press. This work is licensed under a CC BY 4.0 license. Published online: 2022-03-24. The authors are grateful to Graham Cox, Asma Hassannezhad, Konstantin Pankrashkin, David Sher, and Alexander Strohmaier for helpful discussions. Alexandre Girouard and Iosif Polterovich would also like to thank Yakar Kannai for providing them with a copy of the original Hörmander’s manuscript before it was published as [21]. The research of Alexandre Girouard and Iosif Polterovich is partially supported by NSERC, as well as by FRQNT team grant #283055. Mikhail Karpukhin is partially supported by NSF grant DMS-1363432.
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Fonds de recherche du Québec - Nature et technologies (FRQNT)283055
Subject Keywords:Dirichlet-to-Neumann map, Laplace–Beltrami operator, Dirichlet eigenvalues, Robin eigenvalues, eigenvalue asymptotics
Issue or Number:1
Classification Code:2020 Mathematics Subject Classification. Primary 58J50; Secondary 35P20
Record Number:CaltechAUTHORS:20220622-620640200
Persistent URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:115237
Deposited By: Tony Diaz
Deposited On:22 Jun 2022 22:36
Last Modified:28 Jun 2022 18:58

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