2008/01/10 by Rafael D. Benguria, Rafael Benguria, Michael Levitin +4
Computer Science · Mathematics · #35P15 #42B10 #52A40 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP) #math.FA #math.SP #msc:35P15 #msc:42B10 #msc:52A40
paper · pdf · doi:10.48550/arxiv.0801.1617
pdflatex; 4 figures; revised and extended
openalex publication_date 2008/01/10 · arxiv created 2009/06/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⊂\Rbbd 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.