Communications in Mathematical Physics

, Volume 104, Issue 1, pp 103–122

Can one hear the dimension of a fractal?

Authors

  • Jean Brossard
    • Institut FourierUniversité de Grenoble 1
  • René Carmona
    • Department of MathematicsUniversity of California at Irvine
Article

DOI: 10.1007/BF01210795

Cite this article as:
Brossard, J. & Carmona, R. Commun.Math. Phys. (1986) 104: 103. doi:10.1007/BF01210795

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.

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© Springer-Verlag 1986