摘要
This paper is divided into two parts. In the first part the authors extend Kac's classical problem to the fractal case, i.e., to ask: Must two isospectral planar domains with fractal boundaries be isometric?It is demonstrated that the answer to this question is no, by constructing a pair of disjoint isospectral planar domains whose boundaries have the same interior Bouligand-Minkowski dimension but are not isometric. In the second part of this paper the authors give the exact two-term asymptotics for the Dirichlet counting functions associated with the examples given here and obtain sharp two sided estimates for the second term of the counting functions. The first result in the second part of the paper shows that the coefficient of the second term is an oscillatory function of λ, which implies that the Weyl-Berry conjecture, for the examples given here, is false. The second result implies that the weaker form of the Weyl-Berry conjecture, for these examples, is true. This in turn means that the interior Bouligand-Minkowski dimension of the examples is a spectral invariant.
