The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript
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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2021
- DOI:
- 10.48550/arXiv.2102.06594
- arXiv:
- arXiv:2102.06594
- Bibcode:
- 2021arXiv210206594G
- Keywords:
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- Mathematics - Spectral Theory;
- Mathematics - Differential Geometry;
- Primary 58J50. Secondary 35P20
- E-Print:
- Minor changes, the dedication added. 24 pages, 2 figures