FRACTAL DRUMS AND THE N-DIMENSIONAL MODIFIED WEYL-BERRY CONJECTURE

In this paper, we study the spectrum of the Dirichlet Laplacian in a b ounded (or, more generally, of finite volume) open set Omega is an ele ment of R(n) (n greater than or equal to 1) with fractal boundary part ial derivative Omega of interior Minkowski dimension delta is an eleme nt of (n - I,nl. By means of the technique of tessellation of domains, we give the exact second term of the asymptotic expansion of the ''co unting function'' N(lambda) (i.e. the number of positive eigenvalues l ess than lambda) as lambda --> + infinity, which is of the form lambda (delta/2) times a negative, bounded and left-continuous function of la mbda. This explains the reason why the modified Weyl-Berry conjecture does not hold generally for n greater than or equal to 2. in addition, we also obtain explicit upper and lower bounds on the second term of N(lambda).