Steklov zeta-invariants and a compactness ...
Type de document :
Compte-rendu et recension critique d'ouvrage
Titre :
Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains
Auteur(s) :
Jollivet, Alexandre [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Sharafutdinov, Vladimir [Auteur]
Sobolev Institute of Mathematics
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Laboratoire Paul Painlevé - UMR 8524 [LPP]
Sharafutdinov, Vladimir [Auteur]
Sobolev Institute of Mathematics
Titre de la revue :
Journal of Functional Analysis
Pagination :
1712-1755
Éditeur :
Elsevier
Date de publication :
2018-10
ISSN :
0022-1236
Mot(s)-clé(s) en anglais :
Steklov spectrum
Dirichlet-to-Neumann operator
zeta function
inverse spectral problem
Dirichlet-to-Neumann operator
zeta function
inverse spectral problem
Discipline(s) HAL :
Mathématiques [math]/Physique mathématique [math-ph]
Résumé en anglais : [en]
The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function $a$ on ...
Lire la suite >The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function $a$ on the unit circle from the spectrum of the operator $a\Lambda$, where $\Lambda$ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by $Z_m(a)={\rm Tr}[(a\Lambda)^{2m}-(aD)^{2m}]$ for every smooth function $a$. In the case of a positive $a$, zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for $Z_m(a)$ in the case of a real function $a$. On using the estimate, we prove the compactness of a Steklov isospectral family of planar domains in the $C^\infty$-topology. We also describe all real functions $a$ satisfying $Z_m(a)=0$.Lire moins >
Lire la suite >The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function $a$ on the unit circle from the spectrum of the operator $a\Lambda$, where $\Lambda$ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by $Z_m(a)={\rm Tr}[(a\Lambda)^{2m}-(aD)^{2m}]$ for every smooth function $a$. In the case of a positive $a$, zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for $Z_m(a)$ in the case of a real function $a$. On using the estimate, we prove the compactness of a Steklov isospectral family of planar domains in the $C^\infty$-topology. We also describe all real functions $a$ satisfying $Z_m(a)=0$.Lire moins >
Langue :
Anglais
Vulgarisation :
Non
Projet ANR :
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