## Abstract

We study the spectral asymptotics of the Dirichlet-to-Neumann operator Λ_{γ} on a multiply connected, bounded, domain Ω ⊂ R^{d}, d ≥ 3, associated with the uniformly elliptic operator L_{γ} = - Σ_{i,j=1}^{d} ∂_{i}γ_{ij}∂_{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 Λ_{γ} = D_{γ} + R_{γ}, where D_{γ} is a direct sum of operators, each of which acts on one boundary component only, and R_{γ} is a smoothing operator. This representation follows from the fact that the γ-harmonic extensions of eigenfunctions of Λ_{γ} 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 Λ_{γ}.

Original language | English |
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Pages (from-to) | 1717-1741 |

Number of pages | 25 |

Journal | Inverse Problems |

Volume | 17 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2001 |

## ASJC Scopus subject areas

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

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