An inverse problem of the wave equation for a general annular drum in R3 with Robin boundary conditions

This paper is an extension to a recent work of E.M.E. Zayed and I.H. Abdel-Halim [Indian J Pure Appl Math, to appear]. The spectral function <(<mu>)over cap>(t) = Sigma (proportional to)(j=1) e(-it mu 1/2), where {mu}(j-1)(infinity) are the eigenvalues of the negative Laplacian in R-3 is Studied for a variety of domains, where -infinity < t < infinity and i = root -1. The dependences of ii(t) on the connectivity of domains and the Robin boundary conditions are analyzed, particular attention is given to a general annular drum in R-3 together with the Robin boundary conditions on its boundary surfaces. Some geometrical properties of Omega (e.g., the volume, the surface area, the mean curvature and the Gaussian curvature) are determined, from complete knowledge of its eigenvalues, by using the asymptotic expansion of ii(t) for small \t \. (C) 2001 Elsevier Science Ltd. All rights reserved.