Higher dimensional inverse problem of the wave equation for a bounded domain with mixed boundary conditions

This paper is an extension to a recent work of Zayed and Abdel-Halim (Chaos, Solitons \& Fractals, to appear; Acta Math Sci, to appear). The spectral function <(<mu>)over cap> = Sigma (infinity)(j=1), exp(-itE(J)(1/2)), where \{E-J\}(J=1)(infinity) are the eigenvalues of the negative Laplacian in R-3, is studied for a variety of domain, where -infinity < t < infinity and i = root -1. The dependences of <(<mu>)over cap>(t) on the connectivity of a domain and the boundary conditions are analyzed. Particular attention is given to a general bounded domain R in R3 With a smooth boundary surface S, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth parts S-J (J = 1,..., n) of S are considered such that U-j=1(n) S-J. 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. (C) 2001 Elsevier Science Ltd. All rights reserved.