John William Turner

Echoes and glimpses of a distant drum In memory of Mark Kac, on the centenary of his birth

Reference 12 Version 1 Date 20/05/2015

To what extent does the spectrum of the laplacian operator on a domain $D$ with prescribed boundary conditions determine its shape ? This paper first retraces the history of this problem, then Kac's approach in terms of a diffusion process with absorbing boundary conditions. It is shown how the restriction to a polygonal boundary for $D$ in this method, which required taking the limit of an infinite number of sides to obtain a smooth one, can be avoided by using the Duhamel method.