Skip to main content
SpringerLink
Log in

Nodal sets of Laplace eigenfunctions under small perturbations

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We study the stability properties of nodal sets of Laplace eigenfunctions on compact manifolds under specific small perturbations. We prove that nodal sets are fairly stable if such perturbations are relatively small, more formally, supported at a sub-wavelength scale. We do not need any generic assumption on the topology of the nodal sets or the simplicity of the Laplace spectrum. As an indirect application, we are able to show that a certain “Payne property” concerning the second nodal set remains stable under controlled perturbations of the domain.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. We use the analyst’s sign convention, that is, \(-\Delta \) is positive semidefinite.

  2. The heuristic that symmetry implies spectral multiplicity seems to be well-known in the community. If the symmetry group of the plate is non-commutative, then some one can bring in some representation theoretic arguments to justify the above mentioned multiplicity. For a detailed proof, see [35].

  3. Here we have used the word “sweepout” in an intuitive sense. However, for linear combinations of eigenfunctions (not necessarily for same eigenvalue), a more precise mathematical formulation exist for this intuitive picture. See [8] for such a formulation.

References

  1. Abraham, R., Robbin, J.: Transversal mappings and flows. Benjamin, New York (1967)

    MATH  Google Scholar 

  2. Alessandrini, G.: Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains. Comment. Math. Helv. 69(1), 142–154 (1994)

    Article  MathSciNet  Google Scholar 

  3. Anné, C.: Perturbation du spectre \(X \setminus TUB^\epsilon Y\) (conditions de Neumann), Séminaire de Théorie Spectrale et Géométrie, No. 4, Année 1985-1986, 17 - 23, Univ. Grenoble I, Saint-Martin-d’Héres (1986)

  4. Anné, C., Colbois, B.: Opérateur de Hodge-Laplace sur des variétés compactes privées d’un nombre fini de boules. J. Funct. Anal. 115(1), 190–211 (1993)

    Article  MathSciNet  Google Scholar 

  5. Anné, C., Colbois, B.: Spectre du laplacien agissant sur les p-formes différentielles et écrasement d’anses. Math. Ann. 303(3), 545–573 (1995)

    Article  MathSciNet  Google Scholar 

  6. Anné, C., Takahashi, J.: Partial collapsing and the spectrum of the Hodge-de Rham operator. Anal. PDE 8(5), 1025–1050 (2015)

    Article  MathSciNet  Google Scholar 

  7. Bando, S., Urakawa, H.: Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds. Tôhoku Math. J. (2) 35(2), 155–172 (1983)

    Article  MathSciNet  Google Scholar 

  8. Beck, T., Becker-Kahn, S., Hanin, B.: Nodal sets of smooth functions with finite vanishing order and \(p\)-sweepouts. Calc. Var. Partial Differ. Equ. 57(5), 13 (2018). Art. 140

    Article  MathSciNet  Google Scholar 

  9. Chavel, I.: Eigenvalues in Riemannian geometry. Including a chapter by Burton Randol. With an appendix by Jozef Dodziuk. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL. xiv+362 pp (1984)

  10. Cheng, S.-Y.: Eigenfunctions and nodal sets. Comment. Math. Helv. 51(1), 43–55 (1976)

    Article  MathSciNet  Google Scholar 

  11. Colin de Verdière, Y.: Sur la multiplicit’e de la première valeur propre non nulle du laplacien. Comment. Math. Helv. 61, 254–270 (1986)

    Article  MathSciNet  Google Scholar 

  12. Enciso, A., Peralta-Salas, D.: Eigenfunctions with prescribed nodal sets. J. Differ. Geom. 101(2), 197–211 (2015)

    Article  MathSciNet  Google Scholar 

  13. Freitas, P.: Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces. Indiana Univ. Math. J. 51, 305–316 (2002)

    Article  MathSciNet  Google Scholar 

  14. Fukaya, K.: Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. Invent. Math. 87, 517–547 (1987)

    Article  MathSciNet  Google Scholar 

  15. Georgiev, B.: On the lower bound of the inner radius of nodal domains. J. Geom. Anal. 29(2), 1546–1554 (2019)

    Article  MathSciNet  Google Scholar 

  16. Georgiev, B., Mukherjee, M.: Nodal geometry, heat diffusion and Brownian motion. Anal. PDE 11(1), 133–148 (2018)

    Article  MathSciNet  Google Scholar 

  17. Georgiev, B., Mukherjee, M.: Some remarks on nodal geometry in the smooth setting. Calc. Var. Partial Differ. Equ. 58(3), 25 (2019). Art. 93

    Article  MathSciNet  Google Scholar 

  18. Grieser, D., Jerison, D.: The size of the first eigenfunction of a convex planar domain. J. Am. Math. Soc. 11(1), 41–72 (1998)

    Article  MathSciNet  Google Scholar 

  19. Hardt, R., Simon, L.: Nodal sets for solutions of elliptic equations. J. Differ. Geom. 30(2), 505–522 (1989)

    Article  MathSciNet  Google Scholar 

  20. Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Nadirashvili, N.: The nodal line of the second eigenfunction of the Laplacian in \(\mathbb{R}^2\) can be closed. Duke Math. J. 90(3), 631–640 (1997)

    Article  MathSciNet  Google Scholar 

  21. Henrot, A., Pierre, M.: Shape variation and optimization. A geometrical analysis. EMS Tracts in Mathematics, 28. European Mathematical Society (EMS), Zürich. xi+365 pp (2018)

  22. Jerison, D.: The diameter of the first nodal line of a convex domain. Ann. Math. (2) 141(1), 1–33 (1995)

    Article  MathSciNet  Google Scholar 

  23. Komendarczyk, R.: On the contact geometry of nodal sets. Trans. Am. Math. Soc. 358(6), 2399–2413 (2006)

    Article  MathSciNet  Google Scholar 

  24. Kiwan, R.: On the nodal set of a second Dirichlet eigenfunction in a doubly connected domain. Ann. Fac. Sci. Toulouse Math. (6) 27(4), 863–873 (2018)

    Article  MathSciNet  Google Scholar 

  25. Lewy, H.: On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere. Comm. PDE 2, 1233–1244 (1977)

    Article  MathSciNet  Google Scholar 

  26. Mangoubi, D.: Local asymmetry and the inner radius of nodal domains. Comm. PDE 33(9), 1611–1621 (2008)

    Article  MathSciNet  Google Scholar 

  27. Mangoubi, D.: On the inner radius of a nodal domain. Can. Math. Bull. 51(2), 249–260 (2008)

    Article  MathSciNet  Google Scholar 

  28. Melas, A.: On the nodal line of the second eigenfunction of the Laplacian in \({\mathbb{R}}^2\). J. Differ. Geom. 35(1), 255–263 (1992)

    Article  MathSciNet  Google Scholar 

  29. Mukherjee, M.: Variation of Laplace spectra of compact “nearly” hyperbolic surfaces. C. R. Math. Acad. Sci. Paris 355(2), 216–221 (2017)

    Article  MathSciNet  Google Scholar 

  30. Ozawa, S.: Singular variation of domains and eigenvalues of the Laplacian. Duke Math. J. 48(4), 767–778 (1981)

    Article  MathSciNet  Google Scholar 

  31. Palais, R.: Natural operations on differential forms. Trans. Am. Math. Soc. 92, 125–141 (1959)

    Article  MathSciNet  Google Scholar 

  32. Rauch, J., Taylor, M.: Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18, 27–59 (1975)

    Article  MathSciNet  Google Scholar 

  33. Takahashi, J.: Collapsing of connected sums and the eigenvalues of the Laplacian. J. Geom. Phys. 40(3–4), 201–208 (2002)

    Article  MathSciNet  Google Scholar 

  34. Takahashi, J.: Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian. Manuscripta Math. 121(2), 191–200 (2006)

    Article  MathSciNet  Google Scholar 

  35. Taylor, M.: Multiple eigenvalues of operators with noncommutative symmetry groups. unpublished notes available at http://mtaylor.web.unc.edu/files/2018/04/gsym.pdf

  36. Uhlenbeck, K.: Generic properties of eigenfunctions. Am. J. Math. 98(4), 1059–1078 (1976)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to IIT Bombay for providing ideal working conditions. The second named author would like to thank the Council of Scientific and Industrial Research, India for funding which supported his research. The authors would like to acknowledge useful conversations with Bogdan Georgiev, Gopala Krishna Srinivasan and Harsha Hutridurga and the first author would like to thank Nitin Nitsure for asking him the question of sand patterns on symmetrical plates mentioned on page 2. Further, the authors are deeply indebted to Junya Takahashi for giving a detailed clarification of certain aspects of Theorem 3.4. Finally, thanks are due to the anonymous Referees, whose many insightful comments enhanced the quality of the paper appreciably.

Author information

Authors and Affiliations

  1. IIT Bombay, Powai, Maharashtra, 400076, India

    Mayukh Mukherjee & Soumyajit Saha

Authors
  1. Mayukh Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Soumyajit Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mayukh Mukherjee.

Ethics declarations

Conflict of interest

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

Additional information

Communicated by Y. Giga.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukherjee, M., Saha, S. Nodal sets of Laplace eigenfunctions under small perturbations. Math. Ann. 383, 475–491 (2022). https://doi.org/10.1007/s00208-021-02144-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-021-02144-3

Search

Navigation