The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 142–193 | Cite as

Pointwise Bounds for Steklov Eigenfunctions

  • Jeffrey GalkowskiEmail author
  • John A. Toth


Let \((\Omega ,g)\) be a compact, real-analytic Riemannian manifold with real-analytic boundary \(\partial \Omega .\) The harmonic extensions of the boundary Dirichlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfunctions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle \(S^*\partial \Omega .\) These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations \(Pu=0\) near the characteristic set \(\{\sigma (P)=0\}\).


Steklov eigenfunctions FBI transform Analytic microlocal analysis Exponential weighted estimates 

Mathematics Subject Classification

35S05 35P20 58J50 



The authors would like to thank Iosif Polterovich and Steve Zelditch for their comments on an earlier version of this paper. Thanks also to Andras Vasy and Maciej Zworski for valuable suggestions. Finally, thanks to the anonymous referee for many helpful suggestions. J.G. is grateful to the National Science Foundation for support under the Mathematical Sciences Postdoctoral Research Fellowship DMS-1502661. The research of J.T. was partially supported by NSERC Discovery Grant # OGP0170280 and an FRQNT Team Grant. J.T. was also supported by the French National Research Agency project Gerasic-ANR- 13-BS01-0007-0.


  1. 1.
    Bellová, K., Lin, F.-H.: Nodal sets of Steklov eigenfunctions. Calc. Var. Partial Differ. Equ. 54(2), 2239–2268 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203, 2nd edn. Birkhäuser Boston, Inc., Boston, MA (2010)Google Scholar
  3. 3.
    Girouard, A., Polterovich, I.: Spectral geometry of the Steklov problem (survey article). J. Spectr. Theory 7(2), 321–359 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Girouard, A., Parnovski, L., Polterovich, I., Sher, D.A.: The Steklov spectrum of surfaces: asymptotics and invariants. Math. Proc. Cambridge Philos. Soc. 157(3), 379–389 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Guillemin, V., Stenzel, M.: Grauert tubes and the homogeneous Monge-Ampère equation. J. Differ. Geom. 34(2), 561–570 (1991)CrossRefzbMATHGoogle Scholar
  6. 6.
    Hislop, P.D., Lutzer, C.V.: Spectral asymptotics of the Dirichlet-to-Neumann map on multiply connected domains in \({{\mathbb{R}}^d}\). Inverse Probl. 17(6), 1717–1741 (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    Helffer, B.,  Sjöstrand, B.: Résonances en limite semi-classique. Mém. Soc. Math. France (N.S.) (24-25), iv+228 (1986)Google Scholar
  8. 8.
    Jin, L.: Semiclassical Cauchy estimates and applications. Trans. Am. Math. Soc. 369(2), 975–995 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lebeau, G.: The complex Poisson kernel on a compact analytic Riemannian manifold. preprint (2013)Google Scholar
  10. 10.
    Leichtnam, E., Golse, F., Stenzel, M.: Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds. Ann. Sci. École Norm. Sup. (4) 29(6), 669–736 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lempert, L., Szőke, R.: Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds. Math. Ann. 290(4), 689–712 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Martinez, A.: Estimates on complex interactions in phase space. Math. Nachr. 167, 203–254 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Martinez, A.: Microlocal exponential estimates and applications to tunneling. In: Microlocal analysis and spectral theory (Lucca, 1996). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 490, pp. 349–376. Kluwer Acad. Publ., Dordrecht (1997)Google Scholar
  14. 14.
    Nakamura, S.: On Martinez’ method of phase space tunneling. Rev. Math. Phys. 7(3), 431–441 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Polterovich, I., Sher, D.A., Toth, J.A.: Nodal length of Steklov Eigenfunctions on real-analytic Riemannian surfaces (2015). arXiv preprint arXiv:1506.07600
  16. 16.
    Shamma, S.E.: Asymptotic behavior of Stekloff eigenvalues and eigenfunctions. SIAM J. Appl. Math. 20, 482–490 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sjöstrand, J.: Singularités analytiques microlocales. Astérisque. 95, vol. 95 of Astérisque, pp. 1–166. Soc. Math. France, Paris (1982)Google Scholar
  18. 18.
    Sjöstrand, J.: Density of resonances for strictly convex analytic obstacles. Can. J. Math. 48(2), 397–447, (1996) With an appendix by M. ZworskiGoogle Scholar
  19. 19.
    Sjöstrand, J., Uhlmann, G.: Local analytic regularity in the linearized Calderón problem. Anal. PDE 9(3), 515–544 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sogge, C.D., Wang, X., Zhu, J.: Lower bounds for interior nodal sets of Steklov eigenfunctions. Proc. Am. Math. Soc. 144(11), 4715–4722 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Taylor, M.E.: Partial differential equations II. In: Qualitative Studies of Linear Equations. Applied Mathematical Sciences, vol. 116, 2nd edn. Springer, New York (2011)Google Scholar
  22. 22.
    Toth, J.: Eigenfunction decay estimates in the quantum completely integrable cas. Duke Math. J. 93(2), 231–255 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yau, S.T.: Survey on partial differential equations in differential geometry. In: Seminar on Differential Geometry. Ann. of Math. Stud., vol. 102, pp. 3–71. Princeton Univ. Press, Princeton (1982)Google Scholar
  24. 24.
    Yau, S.-T.: Open problems in geometry. In: Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990). Proc. Sympos. Pure Math., vol. 54, pp. 1–28. American Mathematical Society, Providence (1993)Google Scholar
  25. 25.
    Zelditch, S.: Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I. In: Spectral Geometry. Proc. Sympos. Pure Math., vol. 84 , pp. 299–339. American Mathematical Society, Providence, RI (2012)Google Scholar
  26. 26.
    Zelditch, S.: Hausdorff measure of nodal sets of analytic Steklov eigenfunctions. Math. Res. Lett. 22(6), 1821–1842 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhu, J.: Doubling property and vanishing order of Steklov eigenfunctions. Commun. Partial Differ. Equ. 40(8), 1498–1520 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhu, J.: Interior nodal sets of Steklov eigenfunctions on surfaces. Anal. PDE 9(4), 859–880 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zworski, M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence, RI (2012)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontréalCanada

Personalised recommendations