Fourier transform, null variety, and Laplacian's eigenvalues

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• 作者：Rafael Benguria ; Michael Levitin ; Leonid Parnovski
• 关键词：Laplacian ; Dirichlet eigenvalues ; Neumann eigenvalues ; Eigenvalue estimates ; Fourier transform ; Characteristic function ; Pompeiu problem ; Schiffer's conjecture ; Convex sets
• 刊名：Journal of Functional Analysis
• 出版年：2009
• 出版时间：1 October 2009
• 年：2009
• 卷：257
• 期：7
• 页码：2088-2123
• 全文大小：307 K

文摘

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 maximised, among all convex balanced domains 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.