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International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 2, Pages 397-402
http://dx.doi.org/10.1155/S0161171297000513

Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt

Received 8 June 1994; Revised 21 March 1996

Copyright © 1997 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The spectral function Θ(t)=i=1exp(tλj), where {λj}j=1 are the eigenvalues of the negative Laplace-Beltrami operator Δ, is studied for a compact Riemannian manifold Ω of dimension “k” with a smooth boundary Ω, where a finite number of piecewise impedance boundary conditions (ni+γi)u=0 on the parts Ωi(i=1,,m) of the boundary Ω can be considered, such that Ω=i=1mΩi, and γi(i=1,,m) are assumed to be smooth functions which are not strictly positive.