The Dirichlet-to-Neumann map, the boundary Laplacian, and H\"ormander's rediscovered manuscript
Spectral Theory
2021-08-20 v3 Differential Geometry
Abstract
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 H\"ormander from the 1950s. We present H\"ormander'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.
Keywords
Cite
@article{arxiv.2102.06594,
title = {The Dirichlet-to-Neumann map, the boundary Laplacian, and H\"ormander's rediscovered manuscript},
author = {Alexandre Girouard and Mikhail Karpukhin and Michael Levitin and Iosif Polterovich},
journal= {arXiv preprint arXiv:2102.06594},
year = {2021}
}
Comments
Minor changes, the dedication added. 24 pages, 2 figures