Non-density of nodal lines in the clamped plate problem

TL;DR

This paper demonstrates that high-energy eigenfunctions of the clamped plate problem can have large regions without zeros, unlike Laplace eigenfunctions, revealing new geometric properties of these eigenfunctions.

Contribution

It introduces the first example of high-frequency clamped plate eigenfunctions with macroscopic nodal voids, contrasting with classical Laplace eigenfunction behavior.

Findings

Existence of high-frequency eigenfunctions with large nodal voids

Deformation of the unit disk can produce eigenfunctions with no zeros in a sizable region

Nodal sets of clamped plate eigenfunctions can be non-dense and contain 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.