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

Abstract

The trace of the wave kernel \( \hat{\mu }{\left( t \right)} = {\sum\nolimits_{\omega = 1}^\infty {\exp {\left( { -{\text{\bf i}}tE^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}_{\omega } } \right)}} },\;{\text{where}}\;{\left\{ {E_{\omega } } \right\}}^{\infty }_{{\omega = 1}} \) are the eigenvalues of the negative Laplacian \( - \nabla ^{2} = - {\sum\nolimits_{k = 1}^3 {{\left( {\frac{\partial } {{\partial x^{k} }}} \right)}} }^{2} \) in the (x1, x2, x3)–space, is studied for a variety of bounded domains, where −∞ < t < ∞ and \({\text{\bf i}} = {\sqrt { - 1} } \). The dependence of \( \hat{\mu } \) (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 Ω in R3 surrounded by simply connected bounded domains Ω j with smooth bounding surfaces Sj (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+kj−1, . . . , kj) of the bounding surfaces S j are considered, such that \( S_{j} = \cup ^{{k_{j} }}_{{i = 1 + k_{{j - 1}} }} S^{ * }_{i} \) , where k0 = 0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion of \( {\hat{\mu }} \) (t) for small |t|.