Higher dimensional inverse problem of the wave equation for a general multi-connected bounded domain with a finite number of smooth mixed boundary conditions

Abstract

The spectral function μ(t)=∑ J=1 ∞ exp(− itμ J 1/2) where { μ J } J=1 ∞ are the eigenvalues of the negative Laplacian − Δ 3=−∑ υ=1 3(∂/∂ x υ ) 2 in the ( x 1, x 2, x 3)-space, is studied for small | t| for a variety of bounded domains, where −∞< t<∞ and i= −1 . The dependences of μ(t) on the connectivity of bounded domains and the boundary conditions are analysed. Particular attention is given to a general multi-connected bounded domain Ω in R 3 together with a finite number of smooth Dirichlet, Neumann and Robin boundary conditions on the smooth boundaries ∂Ω J (J=1,…,m) of the domain Ω. Some geometrical properties of Ω (e.g., the volume, the surface area, the mean curvature and the Gaussian curvature of Ω) are determined from the asymptotic expansions of μ(t) for small | t|.