Illinois Journal of Mathematics

An isoperimetric inequality for an integral operator on flat tori

Braxton Osting, Jeremy Marzuola, and Elena Cherkaev

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We consider a class of Hilbert–Schmidt integral operators with an isotropic, stationary kernel acting on square integrable functions defined on flat tori. For any fixed kernel which is positive and decreasing, we show that among all unit-volume flat tori, the equilateral torus maximizes the operator norm and the Hilbert–Schmidt norm.

Article information

Illinois J. Math., Volume 59, Number 3 (2015), 773-793.

Received: 29 October 2015
Revised: 9 June 2016
First available in Project Euclid: 30 September 2016

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Zentralblatt MATH identifier

Primary: 35P05: General topics in linear spectral theory 45P05: Integral operators [See also 47B38, 47G10] 52B60: Isoperimetric problems for polytopes 58C40: Spectral theory; eigenvalue problems [See also 47J10, 58E07]


Osting, Braxton; Marzuola, Jeremy; Cherkaev, Elena. An isoperimetric inequality for an integral operator on flat tori. Illinois J. Math. 59 (2015), no. 3, 773--793. doi:10.1215/ijm/1475266407.

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