An inverse problem of the three-dimensional wave equation for a general annular vibrating membrane

This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R-3. The asymptotic expansion of the trace of the wave operator (\μ) over cap (t) = {[}GRAPHICS] exp (-it mu(upsilon)(1/2)) for small \textbackslash{}t\textbackslash{} and i root-1, where \{muv\}(v = 1)(infinity), are the eigenvalues of the negative Laplacian - del(2) =- {[}GRAPHICS] (partial derivative/partial derivativexk in the (x(1), x(2), x(3))-space, is studied for an annular vibrating membrane Omega in R-3 together with its smooth inner boundary surface S-1 and its smooth outer boundary surface S-2. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Sj (j = 1,..., m) of S-1 and S-j{*} (j = m + 1,..., n) of S-2 such that S-1= {[}GRAPHICS] and S-2 = {[}GRAPHICS] are considered. The basic problem is to extract information on the geometry of the annular vibrating membrane Omega from complete knowledge of its eigenvalues by analyzing the asymptotic expansions of the spectral function (\μ) over cap (t) for small \textbackslash{}t\textbackslash{}.