Short-time asymptotics of the two-dimensional wave equation for an annular vibrating membrane with applications in the mathematical physics

We study the influence of a finite container on an ideal gas using the wave equation approach. The asymptotic expansion of the trace of the wave kernel (\μ) over cap (t) = Sigma(n=1)(infinity) exp(-itmu(v)(1/2)) for small \textbackslash{}t\textbackslash{} and i = root-1, where \{mu(v)\}(v=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 an annular vibrating membrane Omega in R-2 together with its smooth inner boundary partial derivativeOmega(1) and its smooth outer boundary partial derivativeOmega(2), where a finite number of Dirichlet, Neumann and 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) such that partial derivativeOmega(1) = boolean ORj=1m Gamma(j) and partial derivativeOmega2 = boolean ORj=m+1m Gamma(j) is considered. The basic problem is to extract information on the geometry of the annular vibrating membrane Omega from complete knowledge of its eigenvalues using the wave equation approach by analyzing the asymptotic expansions of the spectral function (\μ) over cap (t) for small \textbackslash{}t\textbackslash{}. Some applications of (\μ) over cap (t) for an ideal gas enclosed in the general annular bounded domain Omega are given. (C) 2003 Elsevier Ltd. All rights reserved.