Convexity properties of the normalized ...
Document type :
Article dans une revue scientifique: Article original
DOI :
Permalink :
Title :
Convexity properties of the normalized Steklov zeta function of a planar domain
Author(s) :
Journal title :
Journal of Inverse and Ill-posed Problems
Publisher :
De Gruyter
Publication date :
2021
ISSN :
0928-0219
HAL domain(s) :
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
Mathématiques [math]/Physique mathématique [math-ph]
Mathématiques [math]/Théorie spectrale [math.SP]
Mathématiques [math]/Physique mathématique [math-ph]
Mathématiques [math]/Théorie spectrale [math.SP]
English abstract : [en]
We consider the zeta function $\zeta_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve of perimeter $2\pi$. We prove that $\zeta_\Omega''(0)\ge ...
Show more >We consider the zeta function $\zeta_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve of perimeter $2\pi$. We prove that $\zeta_\Omega''(0)\ge \zeta_{\D}''(0)$ with equality if and only if $\Omega$ is a disk where $\D$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le-1$ the estimate $\zeta_\Omega''(s)\ge \zeta_{\D}''(s)$ holds with equality if and only if $\Omega$ is a disk. We then bring examples of domains $\Omega$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.Show less >
Show more >We consider the zeta function $\zeta_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve of perimeter $2\pi$. We prove that $\zeta_\Omega''(0)\ge \zeta_{\D}''(0)$ with equality if and only if $\Omega$ is a disk where $\D$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le-1$ the estimate $\zeta_\Omega''(s)\ge \zeta_{\D}''(s)$ holds with equality if and only if $\Omega$ is a disk. We then bring examples of domains $\Omega$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.Show less >
Language :
Anglais
Peer reviewed article :
Oui
Audience :
Internationale
Popular science :
Non
Collections :
Source :
Submission date :
2025-01-24T16:38:31Z
Files
- document
- Open access
- Access the document
- convexity.pdf
- Open access
- Access the document
- 2008.09476
- Open access
- Access the document