Fourier transform, null variety, and Laplacian's eigenvalues

Abstract

We consider a quantity $κ(Ω)$ -- the distance to the origin from the null variety of the Fourier transform of the characteristic function of $Ω$. We conjecture, firstly, that $κ(Ω)$ is maximized, among all convex balanced domains $Ω\subset\Rbb^d$ of a fixed volume, by a ball, and also that $κ(Ω)$ is bounded above by the square root of the second Dirichlet eigenvalue of $Ω$. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between $κ(Ω)$ and the eigenvalues of the Laplacians.

pdflatex; 4 figures; revised and extended