The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary
The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary
复制标题
带边界的完全黎曼流形的狄利克雷到诺依曼映射
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
G. Uhlmann
中科院分区：
文献类型：
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作者：
M. Lassas;Michael Taylor;G. Uhlmann
研究了由调和函数的柯西数据确定具有边界的完备黎曼流形的问题。这个问题出现在电阻抗断层扫描中，其中人们试图从在身体边界上进行的测量中找到给定身体内部的未知电导率。在这里，我们证明了，如果dim M ≥ 3，则可以从边界的非空开子集Γ上给定的所有调和函数的Cauchy数据集重建一个具有紧边界的完备实解析黎曼流形M。我们注意到，对于这个结果，我们不需要假设的拓扑结构的流形以外的连通性，我们也不需要先验知识的所有的PRM。
We study the problem of determining a complete Riemannian manifold with boundary from the Cauchy data of harmonic functions. This problem arises in electrical impedance tomography, where one tries to find an unknown conductivity inside a given body from measurements done on the boundary of the body. Here, we show that one can reconstruct a complete, real-analytic, Riemannian manifold M with compact boundary from the set of Cauchy data, given on a non-empty open subset Γ of the boundary, of all harmonic functions with Dirichlet data supported in Γ, provided dim M ≥ 3. We note that for this result we need no assumption on the topology of the manifold other than connectedness, nor do we need a priori knowledge of all of ∂M .