The 3D inverse problem for waves with fractal and general annular bounded domain with piecewise smooth Robin boundary

Abstract

This paper deals with the very interesting problem of the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R 3. The spectral distribution 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 domains, where −∞< t<∞ and i= −1 . The dependencies of μ(t) on the connectivity of a domain and the boundary conditions are analysed. Particular attention is given to a general annular bounded domain Ω in R 3 with a smooth inner boundary surface S 1 and a smooth outer boundary surface S 2, where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components S k * (k=1,…,m) of S 1 and on the piecewise smooth components S k * (k=m+1,…,n) of S 2 are considered such that S 1=⋃ k=1 m S k * and S 2=⋃ k= m+1 n S k *. 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|.