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Non-density of nodal lines in the clamped plate problem

Alberto Enciso, Josef Greilhuber

TL;DR

This work proves that nodal sets of high-frequency clamped plate eigenfunctions need not be dense, by constructing small, symmetry-preserving domain perturbations of the unit disk that admit a high-frequency clamped-plate eigenfunction which remains nonzero in a disk of radius about $0.44$. The strategy hinges on decomposing eigenfunctions into a Helmholtz component $v$ and a non-oscillatory SP component $w$, then carefully perturbing the domain so that $v$ vanishes to high order at the origin while $w$ remains nonzero there; the resulting eigenfunction $u=w-v$ thus preserves a central sign. A combination of Hadamard-type shape derivatives, precise bounds on Bessel functions, and a bespoke one-parameter perturbation yields explicit control over the leading Fourier–Bessel components, allowing a macroscopic nodal void to persist for arbitrarily large frequencies. This demonstrates a qualitative difference from Laplacian eigenfunctions and provides a framework for probing nodal geometry via domain perturbations, with potential implications for spectral geometry of higher-order operators. $\Delta^2 u = \lambda^2 u$ with clamped boundary conditions and the associated shape-derivative analysis are central to achieving explicit, quantitative radii of nodal voids.

Abstract

We show that, in contrast to the case of Laplace eigenfunctions, the nodal set of high energy eigenfunctions of the clamped plate problem is not necessarily dense, and can in fact exhibit macroscopic "nodal voids". Specifically, we show that there are small deformations of the unit disk admitting a clamped plate eigenfunction of arbitrarily high frequency that does not vanish in a disk of radius 0.44.

Non-density of nodal lines in the clamped plate problem

TL;DR

This work proves that nodal sets of high-frequency clamped plate eigenfunctions need not be dense, by constructing small, symmetry-preserving domain perturbations of the unit disk that admit a high-frequency clamped-plate eigenfunction which remains nonzero in a disk of radius about . The strategy hinges on decomposing eigenfunctions into a Helmholtz component and a non-oscillatory SP component , then carefully perturbing the domain so that vanishes to high order at the origin while remains nonzero there; the resulting eigenfunction thus preserves a central sign. A combination of Hadamard-type shape derivatives, precise bounds on Bessel functions, and a bespoke one-parameter perturbation yields explicit control over the leading Fourier–Bessel components, allowing a macroscopic nodal void to persist for arbitrarily large frequencies. This demonstrates a qualitative difference from Laplacian eigenfunctions and provides a framework for probing nodal geometry via domain perturbations, with potential implications for spectral geometry of higher-order operators. with clamped boundary conditions and the associated shape-derivative analysis are central to achieving explicit, quantitative radii of nodal voids.

Abstract

We show that, in contrast to the case of Laplace eigenfunctions, the nodal set of high energy eigenfunctions of the clamped plate problem is not necessarily dense, and can in fact exhibit macroscopic "nodal voids". Specifically, we show that there are small deformations of the unit disk admitting a clamped plate eigenfunction of arbitrarily high frequency that does not vanish in a disk of radius 0.44.

Paper Structure

This paper contains 14 sections, 11 theorems, 125 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1: Main ideas
  3. Initial observations
  4. Domain perturbations
  5. Shape derivative of the eigenvalue
  6. Shape derivatives of the Helmholtz and SP components
  7. Construction of a suitable deformation
  8. Consequences for the nodal set: Overview of the remaining proof
  9. Uniform bounds on Bessel functions
  10. Estimates for derivatives of eigenvalues and eigenfunctions
  11. Abstract estimates
  12. Concrete estimates.
  13. Constructing a suitable one-parameter family of perturbations
  14. Proof of Theorem 1: Conclusion

Key Result

Theorem 1

There exists an increasing sequence of integers $(N_k)_{k \in \mathbb N}$ and corresponding domains $(D_{k})_{k \in \mathbb N}$ in $\mathbb R^2$, which converge to the unit disk in the smooth topology, such that $D_k$ admits a clamped plate eigenfunction $u_k$ with squared frequency $\lambda_k \geq where $r_\infty = 0.44367\dots$ is the unique solution in $(0,1)$ to the equation

Figures (1)

  • Figure 1: Structure of the remaining proof

Theorems & Definitions (22)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 12 more