Illinois Journal of Mathematics

An isoperimetric inequality for an integral operator on flat tori

Braxton Osting, Jeremy Marzuola, and Elena Cherkaev

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1475266407

Digital Object Identifier
doi:10.1215/ijm/1475266407

Mathematical Reviews number (MathSciNet)
MR3554232

Zentralblatt MATH identifier
1359.58017

Subjects
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]

Citation

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. https://projecteuclid.org/euclid.ijm/1475266407


Export citation

References

  • A. Baernstein \bsuffixII, A minimum problem for heat kernels of flat tori, Extremal Riemann surfaces, Contemp. Math., vol. 201, Amer. Math. Soc., Providence, RI, 1997, pp. 227–243.
  • R. P. Bambah and C. A. Rogers, Covering the plane with convex sets, J. London Math. Soc. s1-27 (1952), no. 3, 304–314.
  • M. Berger, Sur les premiéres valeurs propres des variétés Riemanniennes, Compos. Math. 26 (1973), no. 2, 129–149.
  • K. J. Boeroeczky and B. Csikós, A new version of L. Fejes Tóth's moment theorem, Studia Sci. Math. Hungar. 47 (2010), no. 2, 230–256.
  • D. P. Bourne, M. A. Peletier and F. Theil, Optimality of the triangular lattice for a particle system with Wasserstein interaction, Comm. Math. Phys. 329 (2014), no. 1, 117–140.
  • J. W. S. Cassels, On a problem of Rankin about the Epstein zeta function, Proc. Glasgow Math. Assoc. 4 (1959), 73–80.
  • P. H. Diananda, Notes on two lemmas concerning the Epstein zeta function, Proc. Glasgow Math. Assoc. 6 (1964), 202–204.
  • V. Ennola, A lemma about the Epstein zeta function, Proc. Glasgow Math. Assoc. 6 (1964), 198–201.
  • G. Fejes Tóth, Sum of moments of convex polygons, Acta Math. Acad. Sci. Hung. 24 (1973), no. 3, 417–421.
  • L. Fejes Tóth, \textcyrRaspolozheniya \textcyrna \textcyrploskosti, \textcyr na \textcyrsfere \textcyri \textcyrv \textcyrprostranstve. \textcyrM., \textcyrFizmatlit, 1958.
  • L. Fejes Tóth, On the isoperimetric property of the regular hyperbolic tetrahedra, Magy. Tud. Akad. Mat. Kut. Intéz. Közl. 8 (1963), 53–57.
  • L. Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, 2nd ed., Springer-Verlag, Berlin, 1972.
  • O. Giraud and K. Thas, Hearing shapes of drums: Mathematical and physical aspects of isospectrality, Rev. Modern Phys. 82 (2010), no. 3, 2213.
  • P. M. Gruber, A short analytic proof of Fejes Tóth's theorem on sums of moments, Aequationes Math. 58 (1999), no. 3, 291–295.
  • P. M. Gruber, Convex and discrete geometry, Springer, Berlin, 2007.
  • A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser Verlag, Basel, 2006.
  • M. Imre, Kreislagerungen auf flächen konstanter krümmung, Acta Math. Acad. Sci. Hung. 15 (1964), no. 1, 115–121.
  • C.-Y. Kao, R. Lai and B. Osting, Maximizing Laplace–Beltrami eigenvalues on compact Riemannian surfaces, to appear in ESAIM Control Optim. Calc. Var. DOI:\doiurl10.1051/cocv/2016008.
  • R. S. Laugesen and B. A. Siudeja, Sums of Laplace eigenvalues: Rotations and tight frames in higher dimensions, J. Math. Phys. 52 (2011), no. 9, \bnumber093703.
  • J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Natl. Acad. Sci. USA 51 (1964), no. 4, 542.
  • H. L. Montgomery, Minimal theta functions, Glasg. Math. J. 30 (1988), 75–85.
  • B. Osting and J. L. Marzuola, Spectrally optimized point set configurations, submitted, 2016.
  • G. Pólya and G. Szegő, Isoperimetric inequalities in mathematical physics, Princeton University Press, Princeton, 1951.
  • R. A. Rankin, A minimum problem for the Epstein zeta-function, Proc. Glasgow Math. Assoc. 1 (1953), 149–158.
  • M. Reed and B. Simon, Methods of modern mathematical physics, vol. 4: Analysis of operators, Academic Press, San Diego, 1978.
  • S. Wolpert, The eigenvalue spectrum as moduli for flat tori, Trans. Amer. Math. Soc. 244 (1978), 313–321.