Stability Estimates for Commutativity Properties of the Dirichlet-to-Neumann Operator
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917001414311936 |
|---|---|
| author | Speciel, Romain |
| author_facet | Speciel, Romain |
| contents | The Laplacian $Δ_{\mathbb{S}^{n-1}}$ on the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ has the property that it can explicitly be expressed in terms of $Λ$, the Dirichlet-to-Neumann map of the unit ball, as $Δ_{\mathbb{S}^{n-1}}=Λ^2+(n-2)Λ$. In this paper, we seek to characterize those manifolds for which such an exact relationship holds, and more generally measure the discrepancy of such a relationship holding in terms of geometric data. To this end, we obtain a stability estimate which shows that, for a smoothly bounded domain in $\mathbb{R}^3$, if the commutator $[Λ,Δ_{\mathbb{S}^{n-1}}]$ is small then that domain is itself close to a ball. We then study the case of manifolds conformal to the ball, show that a relationship as above implies a radial metric structure, and discuss stability in this setting. Finally, we provide a modern exposition of Gohberg's lemma, a foundational result in microlocal analysis which we employ as a starting step for our reasoning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08822 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability Estimates for Commutativity Properties of the Dirichlet-to-Neumann Operator Speciel, Romain Analysis of PDEs Differential Geometry Spectral Theory 58J50 (Primary), 35P05 (Secondary) The Laplacian $Δ_{\mathbb{S}^{n-1}}$ on the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ has the property that it can explicitly be expressed in terms of $Λ$, the Dirichlet-to-Neumann map of the unit ball, as $Δ_{\mathbb{S}^{n-1}}=Λ^2+(n-2)Λ$. In this paper, we seek to characterize those manifolds for which such an exact relationship holds, and more generally measure the discrepancy of such a relationship holding in terms of geometric data. To this end, we obtain a stability estimate which shows that, for a smoothly bounded domain in $\mathbb{R}^3$, if the commutator $[Λ,Δ_{\mathbb{S}^{n-1}}]$ is small then that domain is itself close to a ball. We then study the case of manifolds conformal to the ball, show that a relationship as above implies a radial metric structure, and discuss stability in this setting. Finally, we provide a modern exposition of Gohberg's lemma, a foundational result in microlocal analysis which we employ as a starting step for our reasoning. |
| title | Stability Estimates for Commutativity Properties of the Dirichlet-to-Neumann Operator |
| topic | Analysis of PDEs Differential Geometry Spectral Theory 58J50 (Primary), 35P05 (Secondary) |
| url | https://arxiv.org/abs/2510.08822 |