Spectral asymptotics of the Dirichlet-to-Neumann map on multiply connected domains in Rd

P. D. Hislop, C. V. Lutzer

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

We study the spectral asymptotics of the Dirichlet-to-Neumann operator Λγ on a multiply connected, bounded, domain Ω ⊂ Rd, d ≥ 3, associated with the uniformly elliptic operator Lγ = - Σi,j=1diγijj, 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.

Original languageEnglish
Pages (from-to)1717-1741
Number of pages25
JournalInverse Problems
Volume17
Issue number6
DOIs
StatePublished - Dec 2001

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Signal Processing
  • Mathematical Physics
  • Computer Science Applications
  • Applied Mathematics

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