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The Journal of Analysis

, Volume 26, Issue 1, pp 71–102 | Cite as

Shifted lattices and asymptotically optimal ellipses

  • Richard S. Laugesen
  • Shiya Liu
Original Research Paper

Abstract

Translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions, and consider a decreasing concave (or convex) curve in the first quadrant. Construct a family of curves by rescaling in the coordinate directions while preserving area, and identify the curve in the family that encloses the greatest number of the shifted lattice points. We find the limiting shape of this maximizing curve as the area is scaled up towards infinity. The limiting shape depends explicitly on the lattice shift, except that when the shift is too negative, the maximizing curve fails to converge and instead degenerates. Our results handle the p-circle \(x^p+y^p=1\) when \(p>1\) (concave) and also when \(0<p<1\) (convex). The circular case \(p=2\) with shift \(-1/2\) corresponds to minimizing high eigenvalues in a symmetry class for the Laplacian on rectangles, while the straight line case (\(p=1\)) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.

Keywords

Translated lattice Concave curve Convex curve p-ellipse Spectral optimization Dirichlet Laplacian Schrödinger eigenvalues Harmonic oscillator 

Mathematics Subject Classification

Primary 35P15 Secondary 11P21 52C05 

Notes

Acknowledgements

The material in this paper forms part of Shiya Liu’s Ph.D. dissertation at the University of Illinois, Urbana–Champaign [20].

Funding

This study was funded by the Simons Foundation (Grant #429422 to Richard Laugesen) and the University of Illinois Research Board (award RB17002).

Compliance with ethical standards

Human and animal rights

This research did not involve human participants or animals.

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Antunes, P.R.S. 2014. Optimal bilaplacian eigenvalues. SIAM Journal on Control and Optimization 52 (4): 2250–2260.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antunes, P.R.S., and P. Freitas. 2012. Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. Journal of Optimization Theory and Applications 154: 235–257.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Antunes, P.R.S., and P. Freitas. 2013. Optimal spectral rectangles and lattice ellipses. In Proceedings of the Royal Society of London Series A Mathematical, Physical and Engineering Sciences, No. 2150, 20120492Google Scholar
  4. 4.
    Antunes, P.R.S., and P. Freitas. 2016. Optimisation of eigenvalues of the Dirichlet Laplacian with a surface area restriction. Applied Mathematics and Optimization 73 (2): 313–328.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Antunes, P.R.S., and É. Oudet. 2017. Numerical minimization of Dirichlet-Laplacian eigenvalues of four-dimensional geometries. SIAM Journal on Scientific Computing 39 (3): B508–B521.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ariturk, S., and R. S. Laugesen. Optimal stretching for lattice points under convex curves. Portuguese Mathematical. arXiv:1701.03217 (to appear).
  7. 7.
    van den Berg, M., D. Bucur, and K. Gittins. 2016. Maximizing Neumann eigenvalues on rectangles. Bulletin of the London Mathematical Society 48 (5): 877–894.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    van den Berg, M., and K. Gittins. 2017. Minimising Dirichlet eigenvalues on cuboids of unit measure. Mathematika 63 (2): 469–482.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bucur, D., and P. Freitas. 2013. Asymptotic behaviour of optimal spectral planar domains with fixed perimeter. Journal of Mathematical Physics 54 (5): 053504.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Colbois, B., and A. El Soufi. 2014. Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces. Mathematische Zeitschrift 278: 529–549.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Freitas, P. 2017. Asymptotic behaviour of extremal averages of Laplacian eigenvalues. Journal of Statistical Physics 167 (6): 1511–1518.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gittins, K. and S. Larson. Asymptotic behaviour of cuboids optimising Laplacian eigenvalues. Integral Equations Operator Theory. arXiv:1703.10249 (to appear).
  13. 13.
    Henrot, A. (ed.). 2017. Shape Optimization and Spectral Theory. Berlin: De Gruyter Open.zbMATHGoogle Scholar
  14. 14.
    Huxley, M.N. 1996. Area, Lattice Points, and Exponential Sums. London Mathematical Society Monographs. New Series, vol. 13, The Clarendon Press, Oxford University Press, New York, Oxford Science Publications.Google Scholar
  15. 15.
    Huxley, M.N. 2003. Exponential sums and lattice points. III. In: Proceedings of the London Mathematical Society (3) No. 87, 591–609.Google Scholar
  16. 16.
    Krätzel, E. 2000. Analytische Funktionen in der Zahlentheorie. Teubner–Texte zur Mathematik, vol. 139. Stuttgart: B. G. Teubner.Google Scholar
  17. 17.
    Krätzel, E. 2004. Lattice points in planar convex domains. Monatshefte für Mathematik 143: 145–162.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Larson, S. Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains. arXiv:1611.05680.
  19. 19.
    Laugesen, R.S., and S. Liu. Optimal stretching for lattice points and eigenvalues. Ark. Mat., http://www.math.illinois.edu/~laugesen/ (to appear).
  20. 20.
    Liu, S. 2017. Asymptotically optimal shapes for counting lattice points and eigenvalues. Ph.D. dissertation, University of Illinois, Urbana–Champaign. http://hdl.handle.net/2142/98364.
  21. 21.
    Marshall, N.F. Stretching convex domains to capture many lattice points. arXiv:1707.00682.
  22. 22.
    Marshall, N.F., and S. Steinerberger. Triangles capturing many lattice points. arXiv:1706.04170.
  23. 23.
    Oudet, É. 2004. Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM: Control, Optimisation and Calculus of Variations 10: 315–330.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Forum D'Analystes, Chennai 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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