The Dirichlet-to-Neumann map, the boundary Laplacian, and H\"ormander's rediscovered manuscript

Alexandre Girouard; Mikhail Karpukhin; Michael Levitin; Iosif Polterovich

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arXiv · 2102.06594

The Dirichlet-to-Neumann map, the boundary Laplacian, and H\"ormander's rediscovered manuscript

Alexandre Girouard·Mikhail Karpukhin·Michael Levitin·Iosif Polterovich

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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.

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Alexandre Girouard, Mikhail Karpukhin, Michael Levitin, Iosif Polterovich. 2021-02-12. The Dirichlet-to-Neumann map, the boundary Laplacian, and H\"ormander's rediscovered manuscript. https://arxiv.org/abs/2102.06594

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