The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

Abstract

The asymptotic expansion for small |t| of the trace of the wave kernel $$ \widehat{\mu }{\left( t \right)}{\sum\nolimits_{v = 1}^\infty {\exp {\left( { - {\text{i}}t{\kern 1pt} \mu ^{{\frac{1} {2}}}_{v} } \right)}} } $$ , where $$ {\text{i}} = {\sqrt { - 1} } $$ and $$ {\left\{ {\mu _{v} } \right\}}^{\infty }_{{v = 1}} $$ are the eigenvalues of the negative Laplacian $$ - \Delta = - {\sum\nolimits_{\beta = 1}^2 {{\left( {\frac{\partial } {{\partial x^{\beta } }}} \right)}^{2} } } $$ in the (x 2,x 2)-plane, is studied for a multi-connected vibrating membrane Ω in R 2 surrounded by simply connected bounded domains Ω j with smooth boundaries ∂Ω j (j = 1, ..., n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Γ i (i = 1+k j−1, ..., k j ) of the boundaries ∂Ω j are considered, such that $$ \partial \Omega _{j} = {\bigcup\nolimits_{i = 1 + k_{{j - 1}} }^{k_{j} } {\Gamma _{i} } } $$ and k 0 = 0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e. g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel $$ \widehat{\mu }{\left( t \right)} $$ for small |t|.