01-326 Peter D. Hislop, Carl V. Lutzer

Spectral asymptotics of the Dirichlet-to-Neumann map on multiply connected domains in R^d (105K, LaTex 2e) Sep 12, 01

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Abstract. We study the spectral asymptotics of the Dirichlet-to-Neumann operator $\Lambda_\gamma$ on a multiply-connected, bounded, domain in $\R^d$, $d \geq 3$, associated with the uniformly elliptic operator $L_\gamma = - \sum_{i,j=1}^d \partial_i \gamma_{ij} \partial_j$, where $\gamma$ is a smooth, positive-definite, symmetric matrix-valued function on $\Omega$. We prove that the operator is approximately diagonal in the sense that $\Lambda_\gamma = D_\gamma + R_\gamma$, where $D_\gamma$ is a direct sum of operators, each of which acts on one boundary component only, and $R_\gamma$ is a smoothing operator. This representation follows from the fact that the $\gamma$-harmonic extensions of eigenfunctions of $\Lambda_\gamma$ 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 $\Lambda_\gamma$.

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