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

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 19...

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Bibliographic Details
Published inJournal of spectral theory Vol. 12; no. 1; pp. 195 - 225
Main Authors Girouard, Alexandre, Karpukhin, Mikhail, Levitin, Michael, Polterovich, Iosif
Format Journal Article
LanguageEnglish
Published 01.01.2022
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ISSN1664-039X
1664-0403
DOI10.4171/jst/399

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Summary: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.
ISSN:1664-039X
1664-0403
DOI:10.4171/jst/399