Fourier transform, null variety, and Laplacian's eigenvalues

Details Fourier transform, null variety, and Laplacian's eigenvalues Fourier transform, null variety, and Laplacian's eigenvalues

2008-01-10

arxiv.org/abs/0801.1617v3

Mathematics

Spectral Theory

pdflatex; 4 figures; revised and extended

Scientific paper

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.

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