An inverse eigenvalue problem of the wave equation for a multi-connected region in R-2 together with three different types of boundary conditions

The trace of the wave kernel mu(t) = Sigma(omega=1)(infinity) exp(-itE(omega)(1/2)), where \{E-omega\}(omega=1)(infinity), are the eigenvalues of the negative Laplacian -del(2) = -Sigma(k=1)(2)(partial derivative/partial derivativex(k))(2) in the (x(1), x(2))-plane, is studied for a variety of bounded domains, where -infinity < t < infinity and i = root-1. The dependence of mu(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 membrane in R-2 surrounded by simply connected bounded domains Omega(j) with smooth boundaries partial derivativeOmega(j) (j = 1,..., n), where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components k Gamma(i) (i = I + k(j-1),..., k(j)) of the boundaries partial derivativeOmega(j) is considered, such that partial derivativeOmega(j) = boolean ORi=1+kj-1kj Gamma(i) where k(0) = 0. The basic problem is to extract information on the geometry of Omega using the wave equation approach from complete knowledge of its eigenvalues. Some geometrical quantities of Omega (e.g. the area of Omega, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Omega, etc.) are determined from the asymptotic expansion of the trace of the wave kernel p(t) for small \textbackslash{}t\textbackslash{}. (C) 2003 Elsevier Inc. All rights reserved.