On spectral asymptotics for domains with fractal boundaries of cabbage type (Q1269503)

The second term of the asymptotic expansion of the eigenvalue counting function \(N(\lambda)= \#\{\lambda_j: \lambda_j\leq\lambda\}\), where \(\lambda_j\) are the eigenvalues of the Dirichlet problem \[ -\Delta\psi= \lambda_j\psi,\quad x\in\Omega;\quad \psi= 0,\quad x\in\partial\Omega \] is found in a class of ``cabbage type'' domains with fractal boundaries. These ``cabbage type'' domains have a sequence (or several sequences) of cracks which converge to the other boundary of the domain. The paper contains detailed proofs. A survey of results concerning this theme is given also.