Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains

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Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains. / Jollivet, Alexandre; Sharafutdinov, Vladimir.

In: Journal of Functional Analysis, Vol. 275, No. 7, 01.10.2018, p. 1712-1755.

Research output: Contribution to journal › Article › peer-review

Harvard

Jollivet, A & Sharafutdinov, V 2018, 'Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains', Journal of Functional Analysis, vol. 275, no. 7, pp. 1712-1755. https://doi.org/10.1016/j.jfa.2018.06.019

APA

Jollivet, A., & Sharafutdinov, V. (2018). Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains. Journal of Functional Analysis, 275(7), 1712-1755. https://doi.org/10.1016/j.jfa.2018.06.019

Vancouver

Jollivet A, Sharafutdinov V. Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains. Journal of Functional Analysis. 2018 Oct 1;275(7):1712-1755. doi: 10.1016/j.jfa.2018.06.019

Author

Jollivet, Alexandre ; Sharafutdinov, Vladimir. / Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains. In: Journal of Functional Analysis. 2018 ; Vol. 275, No. 7. pp. 1712-1755.

BibTeX

@article{0a7615071c62459a8214c56118988aa2,

title = "Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains",

abstract = "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 aΛ, where Λ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by Zm(a)=Tr[(aΛ)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 Zm(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∞-topology. We also describe all real functions a satisfying Zm(a)=0.",

keywords = "Dirichlet-to-Neumann operator, Inverse spectral problem, Steklov spectrum, Zeta function, SETS, NEUMANN OPERATOR",

author = "Alexandre Jollivet and Vladimir Sharafutdinov",

note = "Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

year = "2018",

month = oct,

day = "1",

doi = "10.1016/j.jfa.2018.06.019",

language = "English",

volume = "275",

pages = "1712--1755",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "7",

}

RIS

TY - JOUR

T1 - Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains

AU - Jollivet, Alexandre

AU - Sharafutdinov, Vladimir

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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 aΛ, where Λ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by Zm(a)=Tr[(aΛ)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 Zm(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∞-topology. We also describe all real functions a satisfying Zm(a)=0.

AB - 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 aΛ, where Λ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by Zm(a)=Tr[(aΛ)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 Zm(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∞-topology. We also describe all real functions a satisfying Zm(a)=0.

KW - Dirichlet-to-Neumann operator

KW - Inverse spectral problem

KW - Steklov spectrum

KW - Zeta function

KW - SETS

KW - NEUMANN OPERATOR

UR - http://www.scopus.com/inward/record.url?scp=85049753622&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2018.06.019

DO - 10.1016/j.jfa.2018.06.019

M3 - Article

AN - SCOPUS:85049753622

VL - 275

SP - 1712

EP - 1755

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 7

ER -

ID: 14481889