Can one hear the dimension of a fractal?
type: Item
instance of: scholarly article
title: Can one hear the dimension of a fractal?
zbMATH Open document ID: 0607.58043
DOI: 10.1007/BF01210795
author: Jean Brossard
author: René Carmona
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: We consider the spectrum of the Laplacian in a bounded open domain of \({\mathbb{R}}^ n\) with a rough boundary (i.e. with possibly non-integer dimension) and we discuss a conjecture by M. V. Berry generalizing Weyl's conjecture. Then using ideas Mark Kac developed in his famous study of the drum, we give upper and lower bounds for the second term of the expansion of the partition function. The main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on Minkowski dimensions and on Minkowski measures rather than on Hausdorff ones.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 3982138
zbMATH Keywords: Laplacian
zbMATH Keywords: partition function
zbMATH Keywords: Minkowski dimensions
zbMATH Keywords: Minkowski measures
MaRDI profile type: Publication
Wikidata QID: Q56455397
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full work available at URL: https://doi.org/10.1007/bf01210795
OpenAlex ID: W2054248245
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description: scientific article; zbMATH DE number 3982138