Can one hear the area of a drum?

Sammanfattning

This thesis focuses on proving Weyl’s law for arbitrary, two-dimensional domains. The title is inspired by the expression popularized by Mark Kac in his article ”Can one hear the shape of a drum?” The considered domains are characterized such that they resemble the physical structure of membranes with clamped endpoints, where the boundary can attain any shape as long as it is smooth. By considering the wave equation for the perpendicular displacements of any point of a domain, it is derived that the constituted homogeneous Dirichlet eigenvalue problem yields a spectra of eigenvalues that is correlated to the domain’s vibrational frequencies. Consequently, some of the classical results of the Laplacian and the related eigenfunctions and eigenvalues are highlighted and then utilized in the context of variational calculus to find upper and lower bounds to spectra of different shapes. The homogeneous Dirichlet and Neumann eigenvalue problems are also solved for some particular shapes known to have explicit solutions. These results are then used to approximate arbitrary domains and to lastly prove Weyl’s law, which describes an asymptotic relationship between the spectra and the area of a domain. While the result can be extended to other dimensions, this thesis only answers the question in the affirmative for the two-dimensional case.