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A metric Sturm–Liouville theory in two dimensions

  • Stefan SteinerbergerEmail author
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Abstract

A central result of Sturm–Liouville theory (also called the Sturm–Hurwitz theorem) states that if \(\phi _k\) is a sequence of eigenfunctions of a second order differential operator on the interval \(I \subset \mathbb {R}\), then any linear combination satisfies a uniform bound on the roots
$$\begin{aligned} \# \left\{ x \in I:\sum _{k \ge n}{ a_k \phi _k(x)} = 0 \right\} \ge n-1. \end{aligned}$$
We provide a sharp (up to logarithmic factors) generalization to two dimensions: let (Mg) be a compact two-dimensional manifold (with or without boundary), let \((\phi _k)\) denote the sequence of eigenfunctions of a uniformly elliptic operator \(-\text{ div }(a(\cdot ) \nabla )\) (with Dirichlet or Neumann boundary conditions). Then, for any linear combination of eigenfunctions above a certain index n,
$$\begin{aligned} f = \sum _{k \ge n}{a_k \phi _k} ~ \text{ we } \text{ have } \quad \mathcal {H}^1 \left\{ x: f(x) = 0\right\} \gtrsim _{} \frac{\sqrt{n}}{\sqrt{\log {n}}} \log \left( n \frac{\Vert f\Vert _{L^2(M)}}{\Vert f\Vert _{L^1(M)}} \right) ^{-1/2} \frac{\Vert f\Vert _{L^1(M)}}{\Vert f \Vert _{L^{\infty }(M)}} . \end{aligned}$$
Examples on \(M=\mathbb {T}^2\) and \(M=\mathbb {S}^2\) shows that this is optimal up to the logarithmic factors. The proof is using optimal transport and a new inequality for the Wasserstein metric \(W_p\): if f(x)dx and g(x)dx are two absolutely continuous measures on a two-dimensional domain M with continuous densities and the same total mass, then, for all \(1 \le p <\infty \),
$$\begin{aligned} W_p(f(x)dx, g(x) dx) \cdot \mathcal {H}^1 \left\{ x \in M: f(x) = g(x) \right\} \gtrsim _{M,p} \frac{\Vert f-g\Vert _{L^1(M)}^{1+1/p}}{\Vert f-g\Vert _{L^{\infty }(M)}}. \end{aligned}$$

Mathematics Subject Classification

28A75 34B24 35B05 35P20 49Q20 

Notes

References

  1. 1.
    Aronson, D.: Non-negative solutions of linear parabolic equations. Ann. Sci. Norm. Super. 22, 607–694 (1968)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arnol’d, V. I.: The branched covering \(CP2 \rightarrow S^4\), hyperbolicity and projective topology. (Russian) Sibirsk. Mat. Zh. 29 (1988), no. 5, 36–47, 237; translation in Siberian Math. J. 29 (1988), no. 5, 717–726 (1989)Google Scholar
  3. 3.
    Arnold, V.I.: Topological problems in wave propagation theory and topological economy principle in algebraic geometry, Third Lecture by V. Arnold at the Meeting in the Fields Institute Dedicated to His 60th Birthday, Fields Institute Communications (1997)Google Scholar
  4. 4.
    Arnold, V.: Topological properties of eigenoscillations in mathematical physics. Proc. Steklov Inst. Math. 273, 25–34 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Aurich, R., Bäcker, A., Schubert, R., Taglieber, M.: Maximum norms of chaotic quantum eigenstates and random waves. Phys. D 129(1–2), 1–14 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bérard, P.: Inégalités isopérimétriques et applications. Domaines nodaux des fonctions propres. Goulaouic–Meyer–Schwartz Seminar, 1981/1982, Exp. No. XI, 10 pp., Ecole Polytech., Palaiseau, (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bérard, P., Helffer, B.: Sturm’s Theorem on Zeros of Linear Combinations of Eigenfunctions, arXiv:1706.08247
  8. 8.
    Bérard, P., Helffer, B.: Sturm’s theorem on the zeros of sums of eigenfunctions: Gelfand’s strategy implemented, arXiv:1807.03990
  9. 9.
    Bérard, P., Helffer, B.: Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle. Lett. Math. Phys. 106, 1729–1789 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Besicovitch, A.S.: On two problems of Loewner. J. Lond. Math. Soc. 27, 141–144 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bôcher, M.: Lecons sur les methodes de Sturm dans la theorie des equations differentielles lineaires et leurs developpements modernes. Gauthier-Villars et Cie, Éditeurs, Paris (1917)zbMATHGoogle Scholar
  12. 12.
    Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs. Ann. Math. 178(2), 443–452 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bondarenko, A., Radchenko, D., Viazovska, M.: Well-separated spherical designs. Constr. Approx. 41(1), 93–112 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bourgain, J.: On Pleijel’s nodal domain theorem. IMRN 13, 1–7 (2013)Google Scholar
  15. 15.
    Brüning, J.: Über Knoten Eigenfunktionen des Laplace-Beltrami Operators. Math. Z. 158, 15–21 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chanillo, S., Muckenhoupt, B.: Nodal geometry on Riemannian manifolds. J. Differ. Geom. 34, 85–91 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Colding, T.H., Minicozzi II, W.P.: Lower bounds for nodal sets of eigenfunctions. Commun. Math. Phys. 306, 777–784 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Courant, R.: Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke. Nachr. Ges. Göttingen 1, 81–4 (1923)zbMATHGoogle Scholar
  19. 19.
    Dobrusin, R.: Definition of a system of random variables by means of conditional distributions. Teor. Verojatnost. i Primenen. 15, 469–497 (1970)MathSciNetGoogle Scholar
  20. 20.
    Dong, R.T.: Nodal sets of eigenfunctions on Riemann surfaces. J. Differ. Geom. 36(2), 493–506 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Donnelly, H.: Nodal sets for sums of eigenfunctions on Riemannian manifolds. Proc. Am. Math. Soc. 121(3), 967–973 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Donnelly, H., Fefferman, C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93(1), 161–183 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Donnelly, H., Fefferman, C.: Nodal sets for eigenfunctions of the Laplacian on surfaces. J. Am. Math. Soc. 3, 332–353 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Eremenko, A., Novikov, D.: Oscillation of Fourier integrals with a spectral gap. J. Math. Pures Appl. (9) 83(3), 313–365 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Eremenko, A., Novikov, D.: Oscillation of functions with a spectral gap. PNAS 101, 5872–5873 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Galaktionov, V., Harwin, P.: Sturm’s theorems on zero sets in nonlinear parabolic equations. In: Amrein, W.O., Hinz, A.M., Pearson, D.P. (eds.) Sturm-Liouville Theory, pp. 173–199. Birkhäuser, Basel (2005)CrossRefGoogle Scholar
  27. 27.
    Han, Q., Lin, F.-H.: On the geometric measure of nodal sets of solutions. J. Partial Differ. Equ. 7(2), 111–131 (1994)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Han, Q., Hardt, R., Lin, F.-H.: Geometric measure of singular sets of elliptic equations. Commun. Pure Appl. Math. 51(11–12), 1425–1443 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Hardt, R., Simon, L.: Nodal sets for solutions of elliptic equations. J. Differ. Geom. 30, 505–522 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Herrmann, H.: Beiträge zur Theorie der Eigenwerten und Eigenfunktionen, Göttingen Dissertation (1932)Google Scholar
  31. 31.
    Hurwitz, A.: Über die Fourierschen Konstanten integrierbarer Funktionen. Math. Ann. 57, 425–446 (1903)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Hezari, H., Wang, Z.: Lower bounds for volumes of nodal sets: an improvement of a result of Sogge–Zelditch. Spectral geometry. In: Proceedings of Symposia in Pure Mathematics, vol. 84, pp. 229–235. American Mathematical Society, Providence, RI (2012)Google Scholar
  33. 33.
    Hezari, H., Sogge, C.: A natural lower bound for the size of nodal sets. Anal. PDE 5(5), 1133–1137 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Jerison, D., Lebeau, G.: Nodal sets of sums of eigenfunctions. In: Christ, M., Kenig, C., Sadosky, C. (eds.) Harmonic Analysis and Partial Differential Equations, Chicago Lectures in Mathematics, pp. 223–239. University of Chicago Press, ChicagoGoogle Scholar
  35. 35.
    Jung, J.: Bounding the number of nodal domains of eigenfunctions without singular points on the square. Israel J. Math. arXiv:1712.09457 (to appear)
  36. 36.
    Lin, F.-H.: Nodal sets of solutions of elliptic and parabolic equations. Commun. Pure Appl. Math. 44(3), 287–308 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Liouville, J.: Mémoire sur le développement de fonctions ou parties de fonctions en séries dont les divers termes sont assujétis à satisfaire à une même équation différentielle du second ordre, contenant un paramètre variable. J. Math. Pures Appl. 1, 253–265 (1836)Google Scholar
  38. 38.
    Lena, C.: Courant-sharp eigenvalues of a two-dimensional torus. Comptes Rendus Math. 353, 535–539 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Logan, B.: Integrals of high-pass functions. SIAM J. Math. Anal. 15(2), 389–405 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Logunov, A.: Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture. Ann. Math. 187, 241–262 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Logunov, A., Malinnikova, E.: Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three. In: 50 Years with Hardy Spaces: A Tribute to Victor Havin, Operator Theory: Advances and Applications 261, BirkhäuserGoogle Scholar
  42. 42.
    Mangoubi, D.: Local asymmetry and the inner radius of nodal domains. Commun. Partial Differ. Equ. 33, 1611–1621 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Nadirashvilli, N.: Geometry of nodal sets and multiplicity of eigenvalues. Curr. Dev. Math. 1997, 231–235 (1997)CrossRefGoogle Scholar
  44. 44.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Peyre, R.: Non-asymptotic equivalence between \(W^2\) distance and \(\dot{H}^{-1}\), in ESAIM: COCV. 24(4), 1489–1501Google Scholar
  46. 46.
    Pleijel, A.: Remarks on Courant’s nodal line theorem. Commun. Pure Appl. Math. 9, 543–550 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Polterovich, I.: Pleijel’s nodal domain theorem for free membranes. Proc. Am. Math. Soc. 137(3), 1021–1024 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Polya, G.: Qualitatives über Wärmeausgleich. Z. Angew. Math. Mech. 13, 125–128 (1933)zbMATHCrossRefGoogle Scholar
  49. 49.
    Sagiv, A., Steinerberger, S.: Transport and Interface: an Uncertainty Principle for the Wasserstein distance, arXiv:1905.07450
  50. 50.
    Sogge, C., Zelditch, S.: Lower bounds on the Hausdorff measure of nodal sets. Math. Res. Lett. 18, 25–37 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Sogge, C., Zelditch, S.: Lower bounds on the Hausdorff measure of nodal sets II. Math. Res. Lett. 19(6), 1361–1364 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Steinerberger, S.: A geometric uncertainty principle with an application to Pleijel’s estimate. Anna. Henri Poincare 15, 2299–2319 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Steinerberger, S.: Oscillatory functions vanish on a large set. Asian J. Math. (to appear) Google Scholar
  54. 54.
    Steinerberger, S.: Wasserstein Distance, Fourier Series and Applications, arXiv:1803.08011
  55. 55.
    Steinerberger, S.: Quantitative Projections in the Sturm Oscillation Theorem. J. Math. Pures Appl. (to appear) Google Scholar
  56. 56.
    Sturm, C.: Mémoire sur les équations différentielles linéaires du second ordre. J. Math. Pures Appl. 1, 106–186 (1836)Google Scholar
  57. 57.
    Sturm, C.: Mémoire sur une classe d’équations à différences partielles. J. Math. Pures Appl. 1, 373–444 (1836)Google Scholar
  58. 58.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)zbMATHCrossRefGoogle Scholar
  59. 59.
    Vasershtein, L.N.: Markov processes on a countable product space, describing large systems of automata. Problemy Peredachi Informatsii 5(3), 64–73 (1969)MathSciNetGoogle Scholar
  60. 60.
    Yau, S.-T.: Problem section. In: Seminar on Differential Geometry, Annals of Mathematical Studies, vol. 102, pp. 669–706. Princeton (1982)Google Scholar
  61. 61.
    Zettl, A.: Sturm-Liouville Theory. Mathematical Surveys and Monographs. American Mathematical Society, Providence (2010)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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