Letters in Mathematical Physics

, Volume 109, Issue 7, pp 1683–1700 | Cite as

Inequalities for the lowest magnetic Neumann eigenvalue

  • S. FournaisEmail author
  • B. Helffer


We study the ground-state energy of the Neumann magnetic Laplacian on planar domains. For a constant magnetic field, we consider the question whether the disc maximizes this eigenvalue for fixed area. More generally, we discuss old and new bounds obtained on this problem.


Spectral theory Isoperimetric inequalities Magnetic Faber-Krahn inequality 

Mathematics Subject Classification




The discussion on this problem started at a nice meeting in Oberwolfach (December 2014) organized by Bonnaillie–Noël, Kovařík and Pankrashkin. We would like to thank M. van den Berg, D. Bucur, B. Colbois, M. Persson Sundqvist, K. Pankrashkin, N. Popoff and A. Savo for discussions around this problem. We also thank the referee for additional references.


Funding was provided by Det Frie Forskningsråd (Grant No. DFF-4181-00221).

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. 1.
    Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. Lond. Math. Soc. Lecture Note Ser. 273, 95–139 (2000)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bauman, P., Phillips, D., Tang, Q.: Stable nucleation for the Ginzburg–Landau system with an applied magnetic field. Arch. Ration. Mech. Anal. 142, 1–43 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bernoff, A., Sternberg, P.: Onset of superconductivity in decreasing fields for general domains. J. Math. Phys. 39, 1272–1284 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brasco, L., De Philippis, G., Velichkov, B.: Faber-Krahn inequality in sharp quantitative form. Duke Math. J. 104(9), 1777–1831 (2015)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bucur, D.: Personal communication (2017, March)Google Scholar
  6. 6.
    Bucur, D., Giacomini, A.: Faber–Krahn inequalities for the Robin–Laplacian: a free discontinuity approach. Arch. Ration. Mech. Anal. 218, 757–824 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Colbois, B., El Soufi, A., Ilias, S., Savo, A.: Eigenvalues upper bounds for the magnetic operator (2017). ArXiv:1709.09482v1. 27 Sep 2017
  8. 8.
    Colbois, B., Savo, A.: Eigenvalue bounds for the magnetic Laplacian (2016). ArXiv:1611.01930v1
  9. 9.
    Colbois, B., Savo, A.: Lower bounds for the first eigenvalue of the magnetic Laplacian. J. Funct. Anal. 274(10), 2818–2845 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ekholm, T., Kovařík, H., Portmann, F.: Estimates for the lowest eigenvalue of magnetic Laplacians. J. Math. Anal. Appl. 439(1), 330–346 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Erdös, L.: Rayleigh-type isoperimetric inequality with a homogeneous magnetic field. Calc. Var. PDE 4, 283–292 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fournais, S., Helffer, B.: Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian. Ann. Inst. Fourier 56(1), 1–67 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fournais, S., Helffer, B.: Spectral Methods in Surface Superconductivity. Progress in Nonlinear Differential Equations and Their Applications, vol. 77. Birkhäuser, Basel (2010)zbMATHGoogle Scholar
  14. 14.
    Fournais, S., Persson Sundqvist, M.: Lack of diamagnetism and the Little–Parks effect. Commun. Math. Phys. 337(1), 191–224 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Freitas, P., Laugesen, R.S.: From Neumann to Steklov and beyond, via Robin: the Weinberger way. arXiv:1810.07461
  16. 16.
    Helffer, B., Morame, A.: Magnetic bottles in connection with superconductivity. Journal of Functional Analysis 185(2), 604–680 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Helffer, B., Persson Sundqvist, M.: On the semi-classical analysis of the Dirichlet Pauli operator. J. Math. Anal. Appl. 449(1), 138–153 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Helffer, B., Persson Sundqvist, M.: On the semi-classical analysis of the Dirichlet Pauli operator-the non simply connected case. Probl. Math. Anal. J. Math. Sci. 226, 4 (2017)zbMATHGoogle Scholar
  19. 19.
    Howard, R., Treibergs, A.: A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature. Rocky Mt. J. Math. 25, 635–684 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kawohl, B.: Overdetermined problems and the p-Laplacian. Acta Math. Univ. Comen. 76, 77–83 (2007)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Krejcirik, D., Lotoreichik, V.: Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, II: non-convex domains and higher dimensions. arXiv:1707.02269
  22. 22.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001)Google Scholar
  23. 23.
    Lu, K., Pan, X.: Eigenvalue problems of Ginzburg–Landau operator in bounded domains. J. Math. Phys. 40(6), 2647–2670 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pankrashkin, K.: An inequality for the maximum curvature through a geometric flow. Arch. Math. 105, 297–300 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pankrashkin, K., Popoff, N.: Mean curvature bounds and eigenvalues of Robin Laplacians. Calc. Var. 54, 1947–1961 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pestov, G., Ionin, V.: On the largest possible circle embedded in a given closed curve. Dokl. Akad. Nauk SSSR 127, 1170–1172 (1959). (in russian)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Polya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton (1951)zbMATHGoogle Scholar
  28. 28.
    Raymond, N.: Sharp asymptotics for the Neumann Laplacian with variable magnetic field in dimension 2. Ann. Henri Poincaré 10(1), 95–122 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sperb, R.: Maximum Principles and Their Applications. Academic Press, New York (1981)zbMATHGoogle Scholar
  30. 30.
    Szegö, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Ration. Mech. Anal. 3, 343–356 (1954)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Talenti, G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(4), 697–718 (1976)MathSciNetzbMATHGoogle Scholar
  32. 32.
    van den Berg, M., Ferone, V., Nitsch, C., Trombetti, C.: On Polya’s inequality for torsional rigidity and first Dirichlet eigenvalue. Integr. Equ. Oper. Theory 86, 579–600 (2016)CrossRefzbMATHGoogle Scholar
  33. 33.
    Weinberger, H.F.: An isoperimetric inequality for the N-dimensional free membrane problem. J. Ration. Mech. Anal. 5, 633–636 (1956)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Aarhus UniversityAarhus CDenmark
  2. 2.LMJLUniversité de Nantes and CNRSNantes CedexFrance

Personalised recommendations