The 3D inverse problem for waves with fractal and general annular bounded domain with piecewise smooth Robin boundary

This paper deals with the very interesting problem of the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R-3. The spectral distribution <(<mu>)over cap>(t) = Sigma (infinity)(j=1) exp(-it,mu (1/2)(j)), where \{mu\}(J=1)(proportional to) are the eigenvalues of the negative Laplacian -Delta (3) = - Sigma (3)(v=1)(partial derivative/partial derivativex(upsilon))(2) in the (x(1),x(2),x(3))-space, is studied for small \textbackslash{}t \textbackslash{} for a variety of domains, where - infinity, < t < infinity and i = root -1. The dependencies of <(<mu>)over cap>(t) on the connectivity of a domain and the boundary conditions are analysed. Particular attention is given to a general annular bounded domain Omega in R-3 With a smooth inner boundary surface S-t and a smooth outer boundary surface S-2, where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components S{*}(k) (k = 1,...,m) of S-1 and on the piecewise smooth components S{*}(k) (k = m + 1...,n) of S-2 are considered such that S-1 = boolean OR (m)(k=1) S{*}(k) and S-2 = U-k=m+1(n) S{*}(k). Some geometrical properties of Omega (e.g., the volume, the surface area, the mean curvature and the Gaussian curvature of Omega) are determined, from the asymptotic expansions of <(<mu>)over cap>(t) for small \textbackslash{}t \textbackslash{}. (C) 2001 Elsevier Science Ltd. All rights reserved.