On asymptotic formulae of eigenvalue counting functions of Laplacian on an unbounded domain with a fractal boundary (Q2721925)

Let \(\Omega\) be an unbounded domain in \(\mathbb{R}^N\) \((N\geq 2)\) with finite Lebesgue measure. Here is considered the following eigenvalue problem on \(\Omega\). NEWLINE\[NEWLINE-\Delta u=\lambda u\text{ in }\Omega,\quad u=0\text{ on }\Gamma= \partial \Omega.NEWLINE\]NEWLINE The authors are mainly interested in the case, when \(\Gamma\) is considerably irregular like a fractal. The main result in this paper is a partial resolution for the Weyl-Berry conjecture and is stated as follows: NEWLINE\[NEWLINEN(\lambda)= \varphi(\lambda) +O\Bigl(\lambda^{d( \Gamma)\over 2}\Bigr)\text{ as }\lambda \to\infty.NEWLINE\]NEWLINE Here by \(N(\lambda)\) is denoted the number of eigenvalues not exceeding \(\lambda\), \(\varphi(\lambda)\) is the well-known ``Weyl term'', and \(d(\Gamma)\) is of fractal dimension of \(\Gamma= \partial\Omega\).