Can one hear the dimension of a fractal?

Jean Brossard, René Carmona

Research output: Contribution to journalArticle

89 Scopus citations

Abstract

We consider the spectrum of the Laplacian in a bounded open domain of ℝn 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 Haussdorff ones.

Original languageEnglish (US)
Pages (from-to)103-122
Number of pages20
JournalCommunications In Mathematical Physics
Volume104
Issue number1
DOIs
StatePublished - Mar 1 1986
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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