An inverse problem of the wave equation for a-general doubly connected region in R-2 with a finite number of piecewise smooth Robin boundary conditions

The spectral distribution (\&mu;) over cap (t) = Sigma(omega=1)(infinity) exp(-itE(omega)(1/2)), where \{E-omega\}(omega=1)(infinity) are the eigenvalues of the negative Laplacian - Delta = -Sigma(k=1)(2) (partial derivative/partial derivativex(k))(2) in the (x(1), x(2))-plane, is studied for a variety of domains, where -infinity < t < infinity and i = root-1. The dependences of (\&mu;) over cap (t) on the connectivity of a bounded domain and the Robin boundary conditions are analyzed. Particular attention is given to a general annular bounded domain Omega in R-2 with a smooth inner boundary partial derivativeOmega(1) and a smooth outer boundary partial derivativeOmega(2) where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Gamma(J) (J = 1,..., m) of partial derivativeOmega(1) and on the piecewise smooth components Gamma(J) (J = m + 1,..., n) of partial derivativeOmega(2) are considered, such that partial derivativeOmega(1) = boolean ORJ=1m Gamma(J) and partial derivativeOmega(2) = boolean ORJ=m+1n Gamma(J). Some geometrical properties of Omega (e.g., the area of Q, the total lengths of its boundary, the curvature of the boundary, the number of holes of Omega, etc.) are determined, from the asymptotic expansions of (\&mu;) over cap (t) for \textbackslash{}t\textbackslash{} -> 0. (C) 2002 Elsevier Science Inc. All rights reserved.