# A metric Sturm–Liouville theory in two dimensions

• Stefan Steinerberger
Article

## 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

## References

1. 1.
Aronson, D.: Non-negative solutions of linear parabolic equations. Ann. Sci. Norm. Super. 22, 607–694 (1968)
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)
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)
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)
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)
10. 10.
Besicovitch, A.S.: On two problems of Loewner. J. Lond. Math. Soc. 27, 141–144 (1952)
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)
12. 12.
Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs. Ann. Math. 178(2), 443–452 (2013)
13. 13.
Bondarenko, A., Radchenko, D., Viazovska, M.: Well-separated spherical designs. Constr. Approx. 41(1), 93–112 (2015)
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)
16. 16.
Chanillo, S., Muckenhoupt, B.: Nodal geometry on Riemannian manifolds. J. Differ. Geom. 34, 85–91 (1991)
17. 17.
Colding, T.H., Minicozzi II, W.P.: Lower bounds for nodal sets of eigenfunctions. Commun. Math. Phys. 306, 777–784 (2011)
18. 18.
Courant, R.: Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke. Nachr. Ges. Göttingen 1, 81–4 (1923)
19. 19.
Dobrusin, R.: Definition of a system of random variables by means of conditional distributions. Teor. Verojatnost. i Primenen. 15, 469–497 (1970)
20. 20.
Dong, R.T.: Nodal sets of eigenfunctions on Riemann surfaces. J. Differ. Geom. 36(2), 493–506 (1992)
21. 21.
Donnelly, H.: Nodal sets for sums of eigenfunctions on Riemannian manifolds. Proc. Am. Math. Soc. 121(3), 967–973 (1994)
22. 22.
Donnelly, H., Fefferman, C.: Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math. 93(1), 161–183 (1988)
23. 23.
Donnelly, H., Fefferman, C.: Nodal sets for eigenfunctions of the Laplacian on surfaces. J. Am. Math. Soc. 3, 332–353 (1990)
24. 24.
Eremenko, A., Novikov, D.: Oscillation of Fourier integrals with a spectral gap. J. Math. Pures Appl. (9) 83(3), 313–365 (2004)
25. 25.
Eremenko, A., Novikov, D.: Oscillation of functions with a spectral gap. PNAS 101, 5872–5873 (2004)
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)
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)
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)
29. 29.
Hardt, R., Simon, L.: Nodal sets for solutions of elliptic equations. J. Differ. Geom. 30, 505–522 (1989)
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)
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)
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)
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)
39. 39.
Logan, B.: Integrals of high-pass functions. SIAM J. Math. Anal. 15(2), 389–405 (1984)
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)
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)
43. 43.
Nadirashvilli, N.: Geometry of nodal sets and multiplicity of eigenvalues. Curr. Dev. Math. 1997, 231–235 (1997)
44. 44.
Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)
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)
47. 47.
Polterovich, I.: Pleijel’s nodal domain theorem for free membranes. Proc. Am. Math. Soc. 137(3), 1021–1024 (2009)
48. 48.
Polya, G.: Qualitatives über Wärmeausgleich. Z. Angew. Math. Mech. 13, 125–128 (1933)
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)
51. 51.
Sogge, C., Zelditch, S.: Lower bounds on the Hausdorff measure of nodal sets II. Math. Res. Lett. 19(6), 1361–1364 (2012)
52. 52.
Steinerberger, S.: A geometric uncertainty principle with an application to Pleijel’s estimate. Anna. Henri Poincare 15, 2299–2319 (2014)
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)
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)
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)