Higher dimensional inverse problem of the wave equation for a bounded domain with mixed boundary conditions

Abstract

This paper is an extension to a recent work of Zayed and Abdel-Halim (Chaos, Solitons & Fractals, to appear; Acta Math Sci, to appear). The spectral function μ(t)=∑ J=1 ∞ exp(− itE J 1/2) , where { E J } J=1 ∞ are the eigenvalues of the negative Laplacian in R 3, is studied for a variety of domain, where −∞< t<∞ and i= −1 . The dependences of μ(t) on the connectivity of a domain and the boundary conditions are analyzed. Particular attention is given to a general bounded domain Ω in R 3 with a smooth boundary surface S, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth parts S J ( J=1,…, n) of S are considered such that S=∪ J=1 n S J . Some geometrical properties of Ω (e.g., the volume, the surface area, the mean curvature and the Gaussian curvature) are determined, from complete knowledge of its eigenvalues.