Skip to main content
SpringerLink
Log in

The Negative Spectrum of the Robin Laplacian

  • Conference paper
  • First Online:
Spectral Theory and Mathematical Physics

Part of the book series: Latin American Mathematics Series ((LAMSUFSC))

  • 370 Accesses

Abstract

In this article we review various results about the negative spectrum of the Laplacian with a mixed attractive Robin boundary condition.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P.R.S. Antunes, P. Freitas, D. Krejčiřík, Bounds and extremal domains for Robin eigenvalues with negative boundary parameter. Adv. Calc. Var. 10(4), 357–379 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Asorey, A.P. Balachandran, J.M. Perez-Pardo, Edge states at phase boundaries and their stability. Rev. Math. Phys. 28(09), 1650020 (2016)

    Google Scholar 

  3. M. Bareket, On an isoperimetric inequality for the first eigenvalue of a boundary value problem. SIAM J. Math. Anal. 8(2), 280–287 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. M.V. Berry, M. Dennis, Boundary-condition-varying circle billiards and gratings: the Dirichlet singularity. J. Phys. A 41(13), 135203, 23 (2008)

    Google Scholar 

  5. V. Bonnaillie, On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners. Asymptot. Anal. 41(3–4), 215–258 (2005)

    MathSciNet  MATH  Google Scholar 

  6. V. Bonnaillie-Noël, M. Dauge, Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners. Ann. Henri Poincaré 7, 899–931 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Bonnaillie-Noël, M. Dauge, N. Popoff, Ground state energy of the magnetic Laplacian on general three-dimensional corner domains. Mémoir. SMF 145, viii+ 138 pp. (2016)

    Google Scholar 

  8. M.-H. Bossel, Membranes élastiquement liées: extension du théorème de Rayleigh-Faber-Krahn et de l’inégalité de Cheeger. C. R. Acad. Sci. Paris Sér. I Math. 302(1), 47–50 (1986)

    MathSciNet  MATH  Google Scholar 

  9. V. Bruneau, N. Popoff, On the negative spectrum of the Robin Laplacian in corner domains. Anal. PDE 9(5), 1259–1283 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Bruneau, K. Pankrashkin, N. Popoff, Eigenvalue counting function for Robin Laplacians on conical domains. J. Geom. Anal. 28(1), 123–151 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Cakoni, N. Chaulet, H. Haddar, On the asymptotics of a Robin eigenvalue problem. C. R. Math. Acad. Sci. Paris 351(13–14), 517–521 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Carron, P. Exner, D. Krejčiřík, Topologically nontrivial quantum layers. J. Math. Phys. 45(2), 774–784 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. E. Colorado, J. García-Melián, The behavior of the principal eigenvalue of a mixed elliptic problem with respect to a parameter. J. Math. Anal. Appl. 377(1), 53–69 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Daners, A Faber-Krahn inequality for Robin problems in any space dimension. Math. Ann. 335(4), 767–785 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Daners, Principal eigenvalues for generalised indefinite Robin problems. Potential Anal. 38(4), 1047–1069 (2013)

    MathSciNet  MATH  Google Scholar 

  16. D. Daners, J.B. Kennedy, On the asymptotic behaviour of the eigenvalues of a Robin problem. Differ. Integr. Equ. 23(7/8), 659–669 (2010)

    MathSciNet  MATH  Google Scholar 

  17. M. Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics, vol. 1341 (Springer, Berlin, 1988)

    Google Scholar 

  18. M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series, vol. 268 (Cambridge University Press, Cambridge, 1999)

    Google Scholar 

  19. P. Exner, A. Minakov, L. Parnovski, Asymptotic eigenvalue estimates for a Robin problem with a large parameter. Port. Math. 71(2), 141–156 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. V. Ferone, C. Nitsch, C. Trombetti, On the maximal mean curvature of a smooth surface. C. R. Math. Acad. Sci. Paris 354(9), 891–895 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. A.V. Filinovskiy, On the asymptotic behavior of eigenvalues and eigenfunctions of the Robin problem with large parameter. Math. Model. Anal. 22(1), 37–51 (2017)

    Article  MathSciNet  Google Scholar 

  22. P. Freitas, D. Krejčiřík, The first Robin eigenvalue with negative boundary parameter. Adv. Math. 280, 322–339 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. V. Georgescu, V. Nistor, On the essential spectrum of N-body Hamiltonians with asymptotically homogeneous interactions. J. Oper. Theory 77(2), 333–376 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. T. Giorgi, R. Smits, Eigenvalue estimates and critical temperature in zero fields for enhanced surface superconductivity. Z. Angew. Math. Phys. 58(2), 224–245 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Grieser, Spectra of graph neighborhoods and scattering. Proc. Lond. Math. Soc. (3) 97(3), 718–752 (2008)

    Google Scholar 

  26. B. Helffer, A. Kachmar, Eigenvalues for the Robin Laplacian in domains with variable curvature. Trans. Am. Math. Soc. 369(5), 3253–3287 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. B. Helffer, K. Pankrashkin, Tunneling between corners for Robin Laplacians. J. Lond. Math. Soc. (2) 91(1), 225–248 (2015)

    Google Scholar 

  28. B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit. I. Commun. Partial Differ. Equ. 9(4), 337–408 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  29. B. Helffer, A. Kachmar, N. Raymond, Tunneling for the Robin Laplacian in smooth planar domains. Commun. Contemp. Math. 19(1), 1650030, 38 (2017)

    Google Scholar 

  30. A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics (Birkhäuser Verlag, Basel, 2006)

    Google Scholar 

  31. M. Khalile, l’Université Paris-Saclay. PhD thesis, Université de Grenoble, 2018

    Google Scholar 

  32. M. Khalile, Spectral asymptotics for Robin Laplacians on polygonal domains. J. Math. Anal. Appl. 461(2), 1498–1543 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Khalile, K. Pankrashkin, Eigenvalues of robin Laplacians in infinite sectors. Math. Nachr. 291(5–6), 928–965 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Khalile, T. Ourmières-Bonafos, K. Pankrashkin, Effective operator for robin eigenvalues in domains with corners. Preprint, arXiv:1809.04998 (2018)

    Google Scholar 

  35. V.A. Kondrat’ev, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16, 209–292 (1967)

    MathSciNet  Google Scholar 

  36. H. Kovařík, On the lowest eigenvalue of Laplace operators with mixed boundary conditions. J. Geom. Anal. 24(3), 1509–1525 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. A.A. Lacey, J.R. Ockendon, J. Sabina, Multidimensional reaction diffusion equations with nonlinear boundary conditions. SIAM J. Appl. Math. 58(5), 1622–1647 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  38. M. Levitin, L. Parnovski, On the principal eigenvalue of a Robin problem with a large parameter. Math. Nachr. 281(2), 272–281 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Y. Lou, M. Zhu, A singularly perturbed linear eigenvalue problem in C 1 domains. Pac. J. Math. 214(2), 323–334 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Marlettta, G. Rozenblum, A Laplace operator with boundary conditions singular at one point. J. Phys. A Math. Theor. 42(12), 125204 (2009)

    Google Scholar 

  41. E. Montevecchi, J.O. Indekeu, Effects of confinement and surface enhancement on superconductivity. Phys. Rev. B 62(21), 14359 (2000)

    Google Scholar 

  42. S.A. Nazarov, N. Popoff, Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition. C. R. Math. Acad. Sci. Paris 356(9), 927–932 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. K. Pankrashkin, On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains. Nanosyst. Phys. Chem. Math. 4(4):474–483 (2013)

    MATH  Google Scholar 

  44. K. Pankrashkin, On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon. Nanosyst. Phys. Chem. Math. 6, 46–56 (2015)

    Article  Google Scholar 

  45. K. Pankrashkin, On the discrete spectrum of Robin Laplacians in conical domains. Math. Model. Nat. Phenom. 11(2), 100–110 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. K. Pankrashkin, N. Popoff, Mean curvature bounds and eigenvalues of Robin Laplacians. Calc. Var. Partial Differ. Equ. 54(2), 1947–1961 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. K. Pankrashkin, N. Popoff, An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter. J. Math. Pures Appl. (9) 106(4), 615–650 (2016)

    Google Scholar 

  48. M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators (Academic [Harcourt Brace Jovanovich Publishers], New York, 1978)

    Google Scholar 

  49. A. Savo, Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds. Preprint, arXiv:1904.07525 (2019)

    Google Scholar 

  50. B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A (N.S.) 38(3), 295–308 (1983)

    Google Scholar 

  51. A.V. Vikulova, Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalities. Preprint, arXiv:1906.11141 (2019)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank the referee who provided useful and detailed comments in order to improve the contents and the presentation of the manuscript.

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux, Talence, Gironde, France

    Nicolas Popoff

Authors
  1. Nicolas Popoff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Popoff .

Editor information

Editors and Affiliations

  1. Departamento de Matemáticas y C. C., Universidad de Santiago de Chile, Santiago, Chile

    Pablo Miranda

  2. Institut de Mathématiques de Bordeaux, Talence, Gironde, France

    Nicolas Popoff

  3. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

    Georgi Raikov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Popoff, N. (2020). The Negative Spectrum of the Robin Laplacian. In: Miranda, P., Popoff, N., Raikov, G. (eds) Spectral Theory and Mathematical Physics. Latin American Mathematics Series(). Springer, Cham. https://doi.org/10.1007/978-3-030-55556-6_12

Download citation

Publish with us

Policies and ethics

Search

Navigation