The APDE seminar on Monday, 4/20, will be given by Josef Greilhuber (Stanford) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Non-density of nodal lines in the clamped plate problem

Abstract: It is well known that the nodal set (i.e., zero set) of an eigenfunction of the Laplacian – modelling a fundamental mode of vibration of an elastic membrane – is dense at the scale of its characteristic wave-length.

In contrast, we show that the nodal set of high energy eigenfunctions of the clamped plate problem – a fourth order PDE modeling a vibrating metal plate – is not necessarily dense and can in fact exhibit macroscopic “nodal voids”.

Specifically, we construct 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.

Remarkably, this radius is sharp, simultaneously providing the asymptotic upper bound for the size of such circular nodal voids among small perturbations of the disk.