The asymptotics of the two-dimensional wave equation for a general multi-connected vibrating membrane with piecewise smooth Robin boundary conditions

The asymptotic expansion for small \textbackslash{}t\textbackslash{} of the trace of the wave kernel (\μ) over cap (t) = Sigma(nu=1)(infinity) exp(-i t mu(nu)(1/2)), where i = root-1 and \{mu(nu)\}(nu=1)(infinity) are the eigenvalues of the negative Laplacian -Delta = -Sigma(beta=1)(2) (partial derivative/partial derivativex(beta))(2) in the (x(1), x(2)) -plane, is studied for a multi-connected vibrating membrane Omega in R-2 surrounded by simply connected bounded domains Omega(j) with smooth boundaries partial derivativeOmega(j) (j = 1,...n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Gamma(i) (i = 1 + k(j-1),..., k(j)) of the boundaries partial derivativeOmega(j) are considered, such that partial derivativeOmega(j) = U-i=1+kj-1(kj) Gamma(i) and k(o) = 0. The basic problem is to extract information on the geometry of Q using the wave equation approach. Some geometric quantities of Omega (e.g. the area of Omega, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Omega, etc.) are determined from the asymptotic expansion of the trace of the wave kernel (\μ) over cap (t) for small \textbackslash{}t\textbackslash{}.