Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture

Abstract

Let $\Omega$ be a bounded open set of ${\mathbb {R}^n}\;(n \geq 1)$ with "fractal" boundary $\Gamma$. We extend Hermann Weyl’s classical theorem by establishing a precise remainder estimate for the asymptotics of the eigenvalues of positive elliptic operators of order $2m\;(m \geq 1)$ on $\Omega$. We consider both Dirichlet and Neumann boundary conditions. Our estimate—which is expressed in terms of the Minkowski rather than the Hausdorff dimension of $\Gamma$—specifies and partially solves the Weyl-Berry conjecture for the eigenvalues of the Laplacian. Berry’s conjecture—which extends to "fractals" Weyl’s conjecture—is closely related to Kac’s question "Can one hear the shape of a drum?"; further, it has significant physical applications, for example to the scattering of waves by "fractal" surfaces or the study of porous media. We also deduce from our results new remainder estimates for the asymptotics of the associated "partition function" (or trace of the heat semigroup). In addition, we provide examples showing that our remainder estimates are sharp in every possible "fractal" (i.e., Minkowski) dimension. The techniques used in this paper belong to the theory of partial differential equations, the calculus of variations, approximation theory and—to a lesser extent—geometric measure theory. An interesting aspect of this work is that it establishes new connections between spectral and "fractal" geometry.