Echoes and glimpses of a distant drum

John Wm. Turner

Abstract. 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.

Keywords: Kac drum problem, inverse methods, diffusion process.

Mathematics Subject Classification: 01A70, 35B30, 35J05, 35P20.

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