We study the spectral asymptotics of the Dirichlet-to-Neumann operator Λγ on a multiply-connected, bounded, domain in R^d, d≥3, associated with the uniformly elliptic operator Lγ = − ∑di,j=1 ∂i γij∂j, where γ is a smooth, positive-definite, symmetric matrix-valued function on Ω. We prove that the operator is approximately diagonal in the sense that Λγ = Dγ + Rγ, where Dγ is a direct sum of operators, each of which acts on one boundary component only, and Rγ is a smoothing operator. This representation follows from the fact that the γ-harmonic extensions of eigenfunctions of Λγ vanish rapidly away from the boundary. Using this representation, we study the inverse problem of determining the number of holes in the body, that is, the number of the connected components of the boundary, by using the high-energy spectral asymptotics of Λγ (Refer to PDF file for exact formulas).
Department, Program, or Center
School of Mathematical Sciences (COS)
Inverse Problems 17N6 (2001) 1717-1741
RIT – Main Campus