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 (mu) over cap (t) exp(-itE(omega)(1/2)) where \{E-omega\}(omega=1)(infinity) are the eigenvalues of the negative Laplacian -del(2) = -Sigma(k=1)(3) in the (x(1), x(2),x(3))-space is studied for a variety of bounded domains, where -infinity < t < infinity and i = root-1 The dependence of (mu) over cap (t) on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating irtembrane Omega in R-3 surrounded by simply connected bounded domains Omega(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 S-i({*}) = (i = 1 + k(j-1),...,k(j)) of the bounding surfaces S (j) are considered, such that S (j) = U (kj)(i = 1+kj-1) S-i({*}), where k(0) = 0. The basic problem is to extract information on the geometry Omega by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Omega (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined A from the asymptotic expansion of (mu) over cap (t) for small vertical bar t vertical bar.