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Echoes and glimpses of a distant drum

Turner John Wm.

Opuscula Mathematica

2020

Vol. 40, no. 6

737-–750

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.