The Laplace operator in fractal domains (Q2707685)

This is a survey based on author's recent papers [``On the eigenfunctions and the first eigenvalue of fractal drums'' (submitted) and ``Function spaces and Dirichlet problems in fractal domains'' (submitted)], the background can be found in the book [``Fractals and spectra related to Fourier analysis and function spaces''. (1997, Zbl 0898.46030)]. NEWLINENEWLINENEWLINELet \(\Omega \) is a bounded \(C^\infty \)-domain in \(\mathbb R^n\) and \(\Gamma \subset \Omega \) be a compact \(d\)-set, \(d<n\). Let \(\mu =\mathcal H/\Gamma \) be the Hausdorff measure restricted to \(\Gamma \). The aim of this paper is to analyse the operator \(-\Delta +\mu \), where \(-\Delta \) is the Dirichlet Laplacian on \(\Omega \), in terms of asymptotic behaviour of eigenvalues of this operator. By physical reasoning this operator reflects the vibration of a drum, given by \(\Omega \), where the mass of the membrane is evenly concentrated in the fractal \(\Gamma \).NEWLINENEWLINEFor the entire collection see [Zbl 0933.00035].