Fourier transform, null variety, and Laplacian's eigenvalues

TL;DR

This paper explores a geometric quantity related to the Fourier transform's null variety and conjectures its maximization by balls, linking it to Laplacian eigenvalues, with partial proofs in two dimensions and for near-circular domains.

Contribution

It introduces a new conjecture connecting the null variety of the Fourier transform to domain shape optimization and Laplacian eigenvalues, providing partial results in specific cases.

Findings

Weaker versions of the conjecture are proved in two dimensions.

The conjecture holds for domains close to a disk.

Links between the null variety quantity and Laplacian eigenvalues are discussed.

Abstract

We consider a quantity $\kappa(\Omega)$ -- the distance to the origin from the null variety of the Fourier transform of the characteristic function of $\Omega$. We conjecture, firstly, that $\kappa(\Omega)$ is maximized, among all convex balanced domains $\Omega\subset\Rbb^d$ of a fixed volume, by a ball, and also that $\kappa(\Omega)$ is bounded above by the square root of the second Dirichlet eigenvalue of $\Omega$. 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 $\kappa(\Omega)$ and the eigenvalues of the Laplacians.