Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with 2π perimeter and of the unit disk

2020/08/21 by Jollivet, Alexandre
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2008.09476

Abstract

We consider the zeta function ζ_Ω for the Dirichlet-to-Neumann operator of a simply connected planar domain Ω bounded by a smooth closed curve of perimeter 2π. We prove that ζ_Ω''(0)≥ ζ_\mathbbD''(0) with equality if and only if Ω is a disk where \mathbbD denotes the closed unit disk. We also provide an elementary proof that for a fixed real s satisfying s≤-1 the estimate ζ_Ω''(s)≥ ζ_\mathbbD''(s) holds with equality if and only if Ω is a disk. We then bring examples of domains Ω close to the unit disk where this estimate fails to be extended to the interval (0,2). Other computations related to previous works are also detailed in the remaining part of the text.

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