The wave equation approach to the two-dimensional inverse problem for a general bounded domain with piecewise smooth mixed boundary conditions

Abstract

The spectral distribution μ^(t)=∑ω=1∞exp(-itEω12), where {Eω}ω=1∞ are the eigenvalues of the negative Laplacian -Δ=-∑ν=12(∂∂xν)2 in the (x1, x2)-plane, is studied for a variety of domains, where –∞ <t < ∞ and i=-1. The dependence 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 R2 with a smooth boundary ∂ω, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth parts γj(j = 1, …, n) of ∂ω are considered such that ∂Ω=∪j=1nΓj. Some geometrical properties of ω (e.g., the area of ω, the total lengths of the boundary, the curvature of its boundary, etc.) are determined, from the asymptotic expansions of μ^(t) for |t|→0.