Bounding the number of nodal domains of eigenfunctions without singular points on the square
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BOUNDING THE NUMBER OF NODAL DOMAINS OF EIGENFUNCTIONS WITHOUT SINGULAR POINTS ON THE SQUARE BY
Junehyuk Jung∗ Department of Mathematics, Texas A&M University College Station, TX 77840, USA e-mail: [email protected]
ABSTRACT
We prove Polterovich’s conjecture concerning the growth of the number of nodal domains for eigenfunctions on a unit square domain, under the assumption that the eigenfunctions do not have any singular points.
1. Introduction Let Ω ⊂ R2 be a domain with a piecewise smooth boundary. Let {φj }j≥1 be an orthonormal Dirichlet eigenbasis with the eigenvalues 0 < λ1 < λ2 ≤ λ3 ≤ · · · , i.e., −Δφj = λj φj φj (x) = 0
for x ∈ ∂Ω,
φj , φk = δjk , where Δ = ∂x2 + ∂y2 is the Laplacian on R2 and δjk is the Kronecker delta. We assume that all φj ’s are real-valued. ∗ We would like to thank Jean Bourgain for introducing the problem, and encourage-
ment. We appreciate Stefan Steinerberger, Van Vu, Igor Wigman, Steve Zelditch, and Joon-Hyeok Yim for many helpful discussions. We also thank Bernard Helffer and the anonymous referee for detailed comments on the earlier version of the manuscript. Received December 26, 2017 and in revised form August 30, 2018
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J. JUNG
Isr. J. Math.
Denoting by N (φj ) the number of nodal domains (connected components of Ω minus the zero set of φj ), Courant’s nodal domain theorem [Cou23] implies that (1.1)
N (φj ) ≤ j
for all j ≥ 1. This inequality is sharp, and we say that λj is a Courant-sharp Dirichlet eigenvalue, whenever the equality is satisfied by φj . For instance, λ1 , λ2 are Courant-sharp Dirichlet eigenvalues. For large values of j, (1.1) is not sharp anymore. In particular, Pleijel’s theorem [Ple56] states that N (φj ) 2 2 (1.2) lim sup = 0.69167 . . . , ≤ j j0,1 j→∞ where j0,1 is the smallest positive zero of the Bessel function, J0 (x). Note that when Ω is a unit square domain D := [0, 1]2 ⊂ R2 , one can prove that 2 N (φj ) ≥ = 0.63661 . . . lim sup j π j→∞ by considering a sequence of Dirichlet eigenfunctions {sin(kπx) sin(kπy)}k=1,2,... . In [Pol09], Polterovich observed that Pleijel’s inequality should not be sharp, because it uses Faber–Krahn inequality which is only sharp on a disk, whereas not all nodal domains can simultaneously be disks. He also conjectured that the maximum number of nodal domains is obtained by the eigenfunctions sin(kπx) sin(kπy) on Ω = D, i.e., (1.3)
lim sup j→∞
N (φj ) 2 ≤ . j π
The idea that not all nodal domains can simultaneously be disks was used by Bourgain [Bou15] to improve Pleijel’s inequality; see also [Ste14]. In [Bou15], the packing density of disks and a refined Faber–Krahn inequality are used to prove that N (φj ) 2 2 lim sup ≤ − 3 · 10−9 . j j0,1 j→∞ Steinerberger [Ste14] utilizes a “geometric uncertainty principle” that quantifies the fact that if a partition is given by sets of equal measure, then the sets cannot
Vol. TBD, 2020
BOUNDING THE NUMBER OF NODAL DOMAINS
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be disks, and proves the existence of a small constant η > 0, such that N (φj ) 2 2 ≤ lim sup − η. j j0,1 j→∞ The main purp
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