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

@inproceedings{Girouard2021TheDM, title={The Dirichlet-to-Neumann map, the boundary Laplacian, and H\"ormander's rediscovered manuscript}, author={Alexandre Girouard and Mikhail A. Karpukhin and Michael Levitin and Iosif Polterovich}, year={2021} }

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… Expand

#### 2 Citations

Nodal count for Dirichlet-to-Neumann operators with potential

- Mathematics
- 2021

We consider Dirichlet-to-Neumann operators associated to ∆+ q on a Lipschitz domain in a smooth manifold, where q is an L∞ potential. We first prove a Courant-type bound for the nodal count of the… Expand

Uniqueness of Yudovich's solutions to the 2D incompressible Euler equation despite the presence of sources and sinks

- Mathematics
- 2021

In 1962, Yudovich proved the existence and uniqueness of classical solutions to the 2D incompressible Euler equations in the case where the fluid occupies a bounded domain with entering and exiting… Expand

#### References

SHOWING 1-10 OF 64 REFERENCES

On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

- Mathematics
- 2012

Abstract We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical… Expand

An inverse spectral result for the Neumann operator on planar domains

- Mathematics
- 1993

Abstract The Neumann operator is an operator on the boundary of a smooth manifold which maps the boundary value of a harmonic function to its normal derivative. In this paper, the Neumann operator on… Expand

SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM

- Mathematics
- 2014

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann… Expand

Heat Invariants of the Steklov Problem

- Mathematics
- 2013

We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and… Expand

On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain

- Mathematics
- 2006

We consider a mixed boundary value problem for a second-order strongly elliptic equation in a Lipschitz domain. The boundary condition on a part of the boundary is of the first order and contains a… Expand

On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator

- Mathematics
- 2005

A simple proof of the inequality µk+1 1. Let Ω be a domain in R d such that the Sobolev space W 1 2 (Ω) is compactly embedded in L2(Ω). Then the spectra of the Dirichlet problem and the Neumann… Expand

Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds

- Mathematics
- 2017

We prove that in Riemannian manifolds the $k$-th Steklov eigenvalue on a domain and the square root of the $k$-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the… Expand

Shape optimization for low Neumann and Steklov eigenvalues

- Mathematics
- 2008

We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature:… Expand

On the principal eigenvalue of a Robin problem with a large parameter

- Mathematics
- 2004

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth… Expand

On the Asymptotic Behavior of the First Eigenvalue of Robin Problem With Large Parameter

- Mathematics
- 2015

We consider the eigenvalue problem Δu + λu = 0 in Ω with Robin condition $$\frac{{\partial u}}{{\partial v}} + \alpha u = 0$$∂u∂v+αu=0 on ∂Q where Ω ⊂ Rsun, n ≥ 2, is a bounded domain with a smooth… Expand