Can one hear the shape of a drum?

Abstract

[en] In this work we study Mark Kac’s classical problem “Can one hear the shape of a drum?” and some of its extensions. They are all inverse problems on characterizing the shape, or at least some geometrical information about the shape, of an Euclidean domain from its Dirichlet spectrum. As to the original problem, we answer it negatively by providing an example of two different shaped planar drums that have the same spectrum of frequencies. As to the extensions, we prove that the spectrum of frequencies of a planar drum characterizes its area. These results are straightforwardly generalized to higher dimensions. Finally, we comment variants of Kac’s problem for which there are positive results for the characterization of the shape of a drum from its spectrum.