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)
P D Hislop and C V Lutzer 2001 Inverse Problems 17 1717
RIT – Main Campus
© 2001 IOP Publishing Ltd. All Rights Reserved.
This is an author-created, un-copyedited version of an article accepted for publication/published in Inverse Problems. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/0266-5611/17/6/313
P. Hislop is supported in part by NSF grant DMS-9707049.ISSN:0266-5611
Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in February 2014.