The 3D Inverse Problem of the Wave Equation for a General Multi-connected Vibrating Membrane with a Finite Number of Piecewise Smooth Boundary Conditions

The trace of the wave kernel ^(t)==1exp(\bf itE1/2),where{E}=1 are the eigenvalues of the negative Laplacian 2=3k=1(xk)2 in the (x 1, x 2, x 3)space, is studied for a variety of bounded domains, where > t > and \bf i=1. The dependence of ^ (t) on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multiconnected vibrating membrane in R 3 surrounded by simply connected bounded domains j with smooth bounding surfaces S j (j = 1, . . . , n), where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si (i = 1*k j1 , . . . , k j ) of the bounding surfaces S j are considered, such that Sj=kji=1*kj1Si , where k 0 = 0. The basic problem is to extract information on the geometry by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion of ^ (t) for small |t|.