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 |t| of the trace of the wave kernel (t)v=1exp(it12v), where i=1 and {v}v=1are the eigenvalues of the negative Laplacian =2=1(x)2 in the (x 2,x 2)-plane, is studied for a multi-connected vibrating membrane in R 2 surrounded by simply connected bounded domains j with smooth boundaries j (j = 1, ..., n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components i (i = 1*k j1, ..., k j ) of the boundaries j are considered, such that j=kji=1*kj1i and k 0 = 0. The basic problem is to extract information on the geometry of using the wave equation approach. Some geometric quantities of (e. g. the area of , the total lengths of its boundary, the curvature of its boundary, the number of the holes of , etc.) are determined from the asymptotic expansion of the trace of the wave kernel (t) for small |t|.