Corpus ID: 231918437

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

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References

SHOWING 1-10 OF 64 REFERENCES
On the first eigenvalue of the Dirichlet-to-Neumann operator on forms
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 classicalExpand
An inverse spectral result for the Neumann operator on planar domains
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 onExpand
SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM
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-NeumannExpand
Heat Invariants of the Steklov Problem
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 andExpand
On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain
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 aExpand
On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator
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 NeumannExpand
Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds
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 theExpand
Shape optimization for low Neumann and Steklov eigenvalues
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
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 smoothExpand
On the Asymptotic Behavior of the First Eigenvalue of Robin Problem With Large Parameter
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 smoothExpand
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