Abstract
The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region inR 2 with mixed boundary conditions, from the complete knowledge of the eigenvalues {λ j } ∞ j=1 for the Laplace operator, using the asymptotic expansion of the spectral function Θ(t)=Σ ∞ j=1 exp(−tλ j ) ast→0.
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References
H. P. W. Gottlieb,Hearing the shape of an annular drum, J. Austral. Math. Soc. Ser. B24, 435–438 (1983).
H. P. W. Gottlieb,Eigenvalues of the Laplacian with Neumann boundary conditions, J. Austral. Math. Soc. Ser. B26, 293–309 (1985).
H. P. W. Gottlieb,Eigenvalues of the Laplacian for rectilinear regions, J. Austral. Math. Soc. Ser. B29, 270–281 (1988).
P. Greiner,An asymptotic expansion for the heat equation, Arch. Rational. Mech. Anal.41, 163–218 (1971).
M. Kac,Can one hear the shape of a drum? Amer. Math. Monthly,73, No. 4, part II, 1–23 (1966).
H. P. McKean Jr and I. M. Singer,Curvature and the eigenvalues of the Laplacian, J. Diff. Geom.,1, 43–69 (1967).
A. Plcijel,A study of certain Green's function with applications in the theory of vibrating membranes, Arkiv. Matematik,2, 553–569 (1953).
B. D. Sleeman and E. M. E. Zayed,An inverse eigenvalue problem for a general convex domain, J. Math. Anal. Appl.,94, No. 1, 78–95 (1983).
B. D. Sleeman and E. M. E. Zayed,Trace formulae for the eigenvalues of the Laplacian, J. Applied Math. Phys. (ZAMP),35, 106–115 (1984).
L. Smith,The asymptotics of the heat equation for a boundary value problem, Invent. Math.63, 467–493 (1981).
K. Stewartson and R. T. Waechter,On hearing the shape of a drum: further results, Proc. Camb. Phil. Soc.,69, 353–363 (1971).
E. M. E. Zayed, Ph.D. Thesis, University of Dundee, Scotland (1981).
E. M. E. Zayed,An inverse eigenvalue problem for the Laplace operator, in ordinary and partial differential equations, (Eds. W. N. Everitt and B. D. Sleeman), Springer, Berlin, 694, 718–726 (1982).
E. M. E. Zayed,Eigenvalues of the Laplacian for the third boundary value problem, J. Austral. Math. Soc. Ser. B29, 79–87 (1987).
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Zayed, E.M.E. Heat equation for an arbitrary doubly-connected region inR 2 with mixed boundary conditions. Z. angew. Math. Phys. 40, 339–355 (1989). https://doi.org/10.1007/BF00945010
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DOI: https://doi.org/10.1007/BF00945010