Spectral asymptotics of the Dirichlet-to-Neumann map on multiply connected domains in R d
Spectral asymptotics of the Dirichlet-to-Neumann map on multiply connected domains in R d
复制标题
R d 中多重连通域上 Dirichlet-to-Neumann 映射的谱渐近
DOI：
10.1088/0266-5611/17/6/313
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发表时间：
2001
期刊：
影响因子：
2.1
通讯作者：
Carl Lutzer
中科院分区：
文献类型：
--
作者：
P. Hislop;Carl Lutzer
研究了多连通有界区域Ω <$d，d ≥ 3上Dirichlet-to-Neumann算子Λγ的谱渐近性，其中γ是Ω上的光滑正定对称矩阵值函数，对应一致椭圆算子Lγ = − ∑ i，j = 1d <$i γij <$j.我们证明了该算子在Λγ = Dγ + Rγ的意义下是近似对角的，其中Dγ是算子的直和，每个算子只作用于一个边界分量，Rγ是光滑算子.这种表示是由Λγ的本征函数的γ-调和延拓在远离边界处迅速消失的事实得出的。利用这个表示，我们研究了利用Λγ的高能谱渐近性确定物体中的孔数，即边界的连通分量数的反问题。
We study the spectral asymptotics of the Dirichlet-to-Neumann operator Λγ on a multiply connected, bounded, domain Ω ⊂ d, d ≥ 3, associated with the uniformly elliptic operator Lγ = − ∑ i,j = 1d ∂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.