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Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media and the Weyl-Berry conjecture. (English) Zbl 0830.35094
Sleeman, B. D. (ed.) et al., Ordinary and partial differential equations. Volume IV. Proceedings of the twelfth Dundee conference held at the University of Dundee, UK, June 22-26, 1992. In honour of Professor D. S. Jones. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 289, 126-209 (1993).
This long survey paper centers around the asymptotic distribution of the eigenvalues of the Laplace operator. Let \(\Omega \subset \mathbb{R}^m\) be a bounded open set with smooth boundary \(\partial \Omega\) and consider the Dirichlet problem for the Laplacian, \(\Delta\), in \(\Omega\). It is well- known that the corresponding selfadjoint operator has a discrete spectrum, \(0 < \lambda_1 < \lambda_2 \leq \cdots \leq \lambda_n \to \infty\) (counted with multiplicity) and already in 1920 Richard Courant proved the asymptotic relation \[ N (\lambda) : = \# \bigl\{ j \mid \lambda_j \leq \lambda \bigr\} = C_m \operatorname{vol} \Omega \lambda^{m/2}+O \bigl( \lambda^{(m-1)/2} \log \lambda \bigr), \] where \(C_m\) is a constant depending only on \(m\); the main tool in the proof was the max-min-principle. Employing (considerably more refined) wave techniques, Seeley and Pham The Lai sharpened the \(O\)-term to \(O (\lambda^{(m-1)/2})\) which is best possible in general. Later Ivrii was able to show that the \(O\)-term can even be replaced by \[ D_m \operatorname{vol}_{m-1} \partial \Omega \lambda^{(m-1)/2}+o \bigl( \lambda^{(m- 1)/2}), \] if only a certain (generic) property is satisfied by the billiard map in \(\Omega\).
The present author describes in detail and with great historical accuracy the attempts that have been made in the last 15 years to generalize such results to domains with very irregular, in particular “fractal” boundary, notably in the author’s own work (partially coauthored). Since it is impossible to describe all results here in detail we single out only a few interesting phenomena not visible in the smooth case.
1) The investigations were stirred by the physicist M. Berry who conjectured that the remainder term should be \(O (\lambda^{H/2})\) if \(H\) denotes the Hausdorff dimension of \(\partial \Omega\). This was corrected into \(O (\lambda^{D/2})\), \(D\) the Minkowski dimension, and Lapidus was able to prove this if the upper Minkowski content of \(\partial \Omega\) is finite and \(D \in (m-1, m)\), using refined max-min- techniques. In the smooth case, his result reduces to Courant’s showing that the case \(D = m-1\) is critical. This corresponds to the fact that, in the smooth case, the \(O (\lambda^{(m-1)/2})\)-estimate could not yet be proved using variational techniques.
2) Examples of special cases show that the analogue of Ivrii’s estimate - - if it holds at all in some generality – has to be more complicated i.e. the dependence on \(\partial \Omega\) of the constant multiplying \(\lambda^{D/2}\) cannot be only through Minkowski content.
Besides a number of very interesting examples the author gives an amusing interpretation of the Riemann hypothesis in terms of “fractal spectral geometry”. Finally, first steps are outlined to develop a spectral theory for the “Laplacian” on fractal domains, at least those with “self-similarity”. It is to be hoped that the highly developed techniques of spectral analysis for smooth elliptic operators find suitable analogues in this irregular setting in the course of these investigations.
For the entire collection see [Zbl 0777.00051].

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
28A80 Fractals
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses