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

Girouard, A., Karpukhin, M., Levitin, M. and Polterovich, I. (2021) The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript. Journal of Spectral Theory. ISSN 1664-0403 (In Press)

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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
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:100049
Uncontrolled Keywords:Dirichlet-to-Neumann map, Laplace–Beltrami operator, Dirichlet eigenvalues, Robin eigenvalues, eigenvalue asymptotics
Publisher:European Mathematical Society

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