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

TL;DR

This paper explores the relationship between the Dirichlet-to-Neumann map and the boundary Laplacian, revealing connections to H"ormander's 1950s work and applying these insights to spectral estimates on various boundary operators.

Contribution

It rediscoveres and applies H"ormander's approach to analyze DtN maps, extending results to non-smooth boundaries, Helmholtz equations, and differential forms.

Findings

Eigenvalue estimates for DtN maps on non-smooth boundaries

Spectral asymptotics for boundary operators

Applications to Helmholtz and differential form boundary problems

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.