Can one hear the area of a drum?

Abstract

Timbre is one of the most important treats in music as it enables the distinction of different instruments even when each one plays the same note. For more general vibrating smooth bounded domains, timbre can be related with the set of eigenvalues of the Laplacian. Having this considered, the aim of this work is to explore this vibrational problem, by studying the existence of solutions of the Laplacian problem with different boundary conditions and explore the inverse spectral problem with Weyl’s law, in order to see how the spectrum determines the area or volume of the domain. In addition, nodal domains will be defined and Courant’s theorem proved and sharpened.