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On hearing the shape of a drum: an extension to higher dimensions

Published online by Cambridge University Press:  24 October 2008

R. T. Waechter
R. T. Waechter
Affiliation:
Department of Mathematics, University College London, Gower Street, London, W.C.1
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Extract

The inverse eigenvalue problem for vibrating membranes (4), may also be examined in three or more dimensions. Let us suppose that λn are the eigen values of the problem

where Ω is a closed convex region or body in En and S is the bounding surface of Ω. The basic problem is to determine the precise shape of Ω on being given the spectrum of eigenvalues λn. In analogy with the membrane problem, it is clear that the trace function may be constructed in identical fashion; thus

where G(r, r', t) is the Green's function of the diffusion equation

and satisfies the Dirichiet condition G(r, r', t) = 0, r∈S, and the initial condition G(r, r', t) → δ(r–r') as t → 0.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 3 , November 1972 , pp. 439 - 447
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Bonnesen, T. and Fenchel, W.Konvexe Körper, Ergebnisse der Mathematik, Bd. 3 (Springer, Berlin 1934).Google Scholar
(2)Hadwiger, H.Altes und Neues über konvexe Körper (Birkhäuser, Basel, 1955).CrossRefGoogle Scholar
(3)Mckean, H. P. and Singer, I. M.J. Differential Geom. 1 (1967), 43–69.CrossRefGoogle Scholar
(4)Stewartson, K. and Waechter, R. T.Proc. Cambridge Philos. Soc. 69 (1971), 353–363.CrossRefGoogle Scholar