Opuscula Math. 40, no. 6 (2020), 737-750
https://doi.org/10.7494/OpMath.2020.40.6.737

Opuscula Mathematica

# 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.

Full text (pdf)

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• Communicated by Jean Mawhin.
• Revised: 2020-10-14.
• Accepted: 2020-10-16.
• Published online: 2020-12-01.