Skip to main content
SpringerLink
Log in

Lower Bounds for Nodal Sets of Dirichlet and Neumann Eigenfunctions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Let \({\phi}\) be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set \({\{\phi = 0\}}\) .

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

Similar content being viewed by others

References

  1. Brüning J.: Über Knoten Eigenfunktionen des Laplace-Beltrami Operators. Math. Z. 158(1), 15–21 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Colding T.H., Minicozzi W.P.: II. Lower bounds for nodal sets of eigenfunctions. Commun. Math. Phys. 306, 777–784 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Dong R.T.: Nodal sets of eigenfunctions on Riemann surfaces. J. Diff. Geom. 36(2), 493–506 (1992)

    MATH  Google Scholar 

  4. Donnelly H., Fefferman C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93, 161–183 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Donnelly, H., Fefferman, C.: Nodal sets of eigenfunctions: Riemannian manifolds with boundary. In: Analysis, et cetera: Research Papers Published in Honor of Jürgen Moser’s60th Birthday. Ed. P. Rabinowitz, E. Zehdner. Boston, MA: Academic Press, 1990, pp. 251–262

  6. Donnelly H., Fefferman C.: Nodal sets for eigenfunctions of the Laplacian on surfaces. J. Amer. Math. Soc. 3(2), 333–353 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press, 1992

  8. Grieser, D.: L p bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries. Ph. D. Thesis. University of California, Los Angeles: USA, 1992

  9. Han, Q., Lin, F.H.: Nodal Sets of Solutions of Elliptic Differential Equations, online at http://www.nd.edu/~qhan/nodal.pdf

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

    MathSciNet  MATH  Google Scholar 

  11. Hezari, H., Sogge, C.D.: A natural lower bound for the size of nodal sets. http://arxiv.org/abs/1107.3440v3 [math.AP], 2011

  12. Hezari, H., Wang, Z.: Lower bounds for volumes of nodal sets. http://arxiv.org/abs/1107.0092v2 [math.AP], 2011

  13. Hörmander L.: Uniqueness theorems for 2nd-order elliptic operators. Comm. Part. Diff. Eqs. 8, 21–64 (1983)

    Article  MATH  Google Scholar 

  14. Mangoubi D.: A remark on recent lower bounds for nodal sets. Comm. Part. Diff. Eqs. 36(12), 2208–2212 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Savo A.: Lower bounds for the nodal length of eigenfunctions of the Laplacian. Ann. Global Anal. Geom. 19(2), 133–151 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Smith H.F.: Sharp L 2 → L q bounds on spectral projectors for low regularity metrics. Math. Res. Lett. 13(5-6), 967–974 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Smith H.F., Sogge C.D.: On the critical semilinear wave equation outside convex obstacles. J. Amer. Math. Soc. 8(4), 879–916 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Smith H.F., Sogge C.D.: On the L p norm of spectral clusters for compact manifolds with boundary. Acta Math. 198, 107–153 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sogge C.D., Zelditch S.: Lower bounds on the Hausdorff measure of nodal sets. Math. Res. Lett. 18(1), 25–37 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Tataru D.: On the regularity of boundary traces for the wave equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(1), 185–206 (1998)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Sinan Ariturk

Authors
  1. Sinan Ariturk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sinan Ariturk.

Additional information

Communicated by S. Zelditch

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ariturk, S. Lower Bounds for Nodal Sets of Dirichlet and Neumann Eigenfunctions. Commun. Math. Phys. 317, 817–825 (2013). https://doi.org/10.1007/s00220-012-1554-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-012-1554-4

Keywords

Search

Navigation