The Distribution of Eigenfrequencies of Anisotropic Fractal Drums

Motivated by some aspects of boundary value problems for partial differential equations, several authors have recently been concerned with the study of function spaces on and of fractals. We refer mainly to the papers by A. Jonsson and H. Wallin [7–10] and to the book [16] where complete references to this topic are given. Let Ω be a bounded domain in 2# having C ¢ boundary ¦Ω and let 0! d A ! 2. An anisotropic d A -set ΓZΩ having anisotropic deviation 0% a% 1 is, roughly speaking, a compact set which can be covered for any j `. ! with N j C 2jdA rectangles R jl (l ̄ 1,... ,N j ) with volR jl C 2−#j, having sides parallel to the axes and side lengths rj,l " , rj,l # satisfying 2−j("+a) % rj,l # % rj,l " % 2−j("−a)