On spectral asymptotics for domains with fractal boundaries
type: Item
instance of: scholarly article
title: On spectral asymptotics for domains with fractal boundaries
zbMATH Open document ID: 0867.58068
DOI: 10.1007/BF02509797
published in: Communications in Mathematical Physics
publication date: 1997-08-10T00:00:00Z
review text: The authors consider the spectral properties of the Laplacian for domains with fractal boundaries. They determine the second term in the spectral asymptotics for the counting function \(N(\lambda)\) for three classes of fractal boundaries. They show that the Minkowski dimension gives the appropriate answer for cabbage type of boundaries due to the one-dimensional structure of the cabbage type fractal boundaries. The answer is different for bubble type domains and for planar bubble fractals. One of the main points of the present work is to single out fractals for which spectral asymptotics can be found without assuming self-similarity or geometrical conditions which permit separation of variables. The second point is to determine if the order of the second term of spectral asymptotics is always related to the Minkowski dimension of the boundary. There are many other excellent papers on the subject of spectral asymptotics for manifolds with fractal boundaries; in particular it is known that the modified Weyl-Berry conjecture is false. See, for example, 1) \textit{M. van den Bergh}, Proc. Symp. Pure Math. 57, 11-22 (1995; Zbl 0851.35050), 2) \textit{J. Fleckinger, M. Levitin}, and \textit{D. Vassiliev}, Proc. Lond. Math. Soc., III. Ser. 71, No. 2, 372-396 (1995; Zbl 0836.35058), 3) \textit{M. van den Bergh}, Probab. Theory Relat. Fields 100, No. 4, 439-456 (1994; Zbl 0812.60068), 4) \textit{M. Levitin} and \textit{D. Vassiliev}, Proc. Lond. Math. Soc., III. Ser. 72, No. 1, 188-214 (1996; Zbl 0837.35105).
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J37
zbMATH DE Number: 975304
zbMATH Keywords: spectral counting function
zbMATH Keywords: fractal boundary
zbMATH Keywords: Laplacian
zbMATH Keywords: spectral asymptotics
zbMATH Keywords: Weyl-Berry conjecture
author: S. Molchanov
author: B. Vainberg
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02509797
OpenAlex ID: W1996434724
description: scientific article