Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

4 Issues per year


IMPACT FACTOR 2017: 1.676

CiteScore 2017: 1.30

SCImago Journal Rank (SJR) 2017: 2.045
Source Normalized Impact per Paper (SNIP) 2017: 1.138

Mathematical Citation Quotient (MCQ) 2017: 1.15

Online
ISSN
1864-8266
See all formats and pricing
More options …
Volume 10, Issue 4

Issues

Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

Pedro R. S. Antunes
  • Group of Mathematical Physics, Faculdade de Ciências da Universidade de Lisboa, Campo Grande,Edifício C6 1749-016 Lisboa, Portugal
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Pedro Freitas
  • Department of Mathematics, Faculty of Human Kinetics & Group of Mathematical Physics, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edifício C6 1749-016 Lisboa, Portugal
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ David Krejčiřík
  • Corresponding author
  • Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 25068 Řež, Czech Republic
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-08-02 | DOI: https://doi.org/10.1515/acv-2015-0045

Abstract

We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in two and three dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).

Keywords: Eigenvalue optimisation; Robin Laplacian; negative boundary parameter; Bareket’s conjecture

MSC 2010: 58J50; 35P15

References

  • [1]

    M. S. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. Google Scholar

  • [2]

    C. J. S. Alves and P. R. S. Antunes, The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes, Comput. Mater Con. 2 (2005), 251–266. Google Scholar

  • [3]

    C. J. S. Alves and P. R. S. Antunes, The method of fundamental solutions applied to some inverse eigenproblems, SIAM J. Sci. Comput. 35 (2013), A1689–A1708. CrossrefWeb of ScienceGoogle Scholar

  • [4]

    D. E. Amos, Computation of modified Bessel functions and their ratios, Math. Comp. 28 (1974), 239–251. CrossrefGoogle Scholar

  • [5]

    P. R. S. Antunes and P. Freitas, Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians, J. Optim. Theory Appl. 154 (2012), 235–257. Web of ScienceCrossrefGoogle Scholar

  • [6]

    P. R. S. Antunes and P. Freitas, Optimal spectral rectangles and lattice ellipses, Proc. R. Soc. Lond. Ser. A 469 (2013), Article ID 20120492. Google Scholar

  • [7]

    P. R. S. Antunes and P. Freitas, Optimisation of eigenvalues of the Dirichlet Laplacian with a surface area restriction, Appl. Math. Optim. 73 (2016), no. 2, 313–328. CrossrefWeb of ScienceGoogle Scholar

  • [8]

    P. R. S. Antunes, P. Freitas and J. B. Kennedy, Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian, ESAIM Control Optim. Calc. Var. 19 (2013), 438–459. CrossrefWeb of ScienceGoogle Scholar

  • [9]

    M. Bareket, On an isoperimetric inequality for the first eigenvalue of a boundary value problem, SIAM J. Math. Anal. 8 (1977), 280–287. CrossrefGoogle Scholar

  • [10]

    A. Berger, The eigenvalues of the Laplacian with Dirichlet boundary condition in 2 are almost never minimized by disks, Ann. Global Anal. Geom. 47 (2015), 285–304. Google Scholar

  • [11]

    M.-H. Bossel, Membranes élastiquement liées: Extension du théoréme de Rayleigh–Faber–Krahn et de l’inégalité de Cheeger, C. R. Acad. Sci. Paris Sér. I 302 (1986), 47–50. Google Scholar

  • [12]

    D. Bucur and P. Freitas, Asymptotic behaviour of optimal spectral planar domains with fixed perimeter, J. Math. Phys. 54 (2013), Article ID 053504. Web of ScienceGoogle Scholar

  • [13]

    B. Colbois and A. El Soufi, Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces, Math. Z. 278 (2014), 529–546. Web of ScienceCrossrefGoogle Scholar

  • [14]

    D. Daners, A Faber–Krahn inequality for Robin problems in any space dimension, Math. Ann. 335 (2006), 767–785. CrossrefGoogle Scholar

  • [15]

    G. Faber, Beweis dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitz. bayer. Akad. Wiss. 1923 (1923), 169–172. Google Scholar

  • [16]

    V. Ferone, C. Nitsch and C. Trombetti, On a conjectured reversed Faber–Krahn inequality for a Steklov-type Laplacian eigenvalue, Commun. Pure Appl. Anal. 14 (2015), 63–81. Google Scholar

  • [17]

    P. Freitas and D. Krejčiřík, A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains, Proc. Amer. Math. Soc. 136 (2008), 2997–3006. Web of ScienceCrossrefGoogle Scholar

  • [18]

    P. Freitas and D. Krejčiřík, The first Robin eigenvalue with negative boundary parameter, Adv. Math. 280 (2015), 322–339. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    T. Giorgi and R. G. Smits, Monotonicity results for the principal eigenvalue of the generalized Robin problem, Illinois J. Math. 49 (2005), 1133–1143. Google Scholar

  • [20]

    T. Giorgi and R. G. Smits, Eigenvalue estimates and critical temperature in zero fields for enhanced surface superconductivity, Z. Angew. Math. Phys. 58 (2007), 1224–245. Web of ScienceGoogle Scholar

  • [21]

    T. Giorgi and R. G. Smits, Bounds and monotonicity for the generalized Robin problem, Z. Angew. Math. Phys. 59 (2008), 600–618. CrossrefWeb of ScienceGoogle Scholar

  • [22]

    A. Girouard, N. Nadirashvili and I. Polterovich, Maximization of the second positive Neumann eigenvalue for planar domains, J. Diff. Geometry 83 (2009), 637–662. CrossrefGoogle Scholar

  • [23]

    D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, Cambridge University Press, New York, 2005. Google Scholar

  • [24]

    T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966. Google Scholar

  • [25]

    J. B. Kennedy, An isoperimetric inequality for the second eigenvalue of the Laplacian with Robin boundary conditions, Proc. Amer. Math. Soc. 137 (2009), 627–633. Google Scholar

  • [26]

    E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1924), 97–100. Google Scholar

  • [27]

    E. Krahn, Über Minimaleigenschaft der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A 9 (1926), 1–44. Google Scholar

  • [28]

    A. A. Lacey, J. R. Ockendon and J. Sabina, Multidimensional reaction diffusion equations with nonlinear boundary conditions, SIAM J. Appl. Math. 58 (1998), no. 5, 1622–1647. CrossrefGoogle Scholar

  • [29]

    M. Levitin and L. Parnovski, On the principal eigenvalue of a Robin problem with a large parameter, Math. Nachr. 281 (2008), 272–281. CrossrefWeb of ScienceGoogle Scholar

  • [30]

    E. Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain, ESAIM Control Optim. Calc. Var. 10 (2004), 315–330. CrossrefGoogle Scholar

  • [31]

    L. E. Payne and H. F. Weinberger, Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl. 2 (1961), 210–216. CrossrefGoogle Scholar

  • [32]

    G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Ann. of Math. Stud. 27, Princeton University Press, Princeton, 1951. Google Scholar

  • [33]

    J. W. S. Rayleigh, The Theory of Sound, 1st ed., Macmillan, London, 1877. Google Scholar

  • [34]

    J. Segura, Bounds for ratios of modified Bessel functions and associated Turán-type inequalities, J. Math. Anal. Appl. 374 (2011), 516–528. CrossrefGoogle Scholar

  • [35]

    G. Szegő, Inequalities for certain eigenvalues of a membrane of given area, J. Ration. Mech. Anal. 3 (1954), 343–356. Google Scholar

  • [36]

    H. F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem, J. Ration. Mech. Anal. 5 (1956), 633–636. Google Scholar

About the article


Received: 2015-11-09

Accepted: 2016-06-21

Published Online: 2016-08-02

Published in Print: 2017-10-01


Funding Source: Fundação para a Ciência e a Tecnologia

Award identifier / Grant number: PTDC/MAT-CAL/4334/2014

Award identifier / Grant number: IF/00177/2013

This research was partially supported by FCT (Portugal) through project PTDC/MAT-CAL/4334/ 2014. The research of the first author was partially supported by FCT, Portugal, through the program “Investigador FCT” with reference IF/00177/2013. The research of the third author was supported by the project RVO61389005 and the GACR grant No. 14-06818S.


Citation Information: Advances in Calculus of Variations, Volume 10, Issue 4, Pages 357–379, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2015-0045.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Comments (0)

Please log in or register to comment.
Log in