三维共形St{ä}ckel流形上的各向异性Calder{ó}n问题
The anisotropic Calder{ó}n problem on 3-dimensional conformally St{ä}ckel manifolds
摘要：
共形St{ä}ckel流形可以刻画为n维伪黎曼流形（M，G）的一类，在该流形上，零测地线的Hamilton-Jacobi方程G（$\nabla$ u，$\nabla$ u）= 0和Laplace方程-- $Δ$ G $ψ$ = 0可由R-分离变量解.在度量具有黎曼签名的特殊情况下，他们给出了Laplace-Beltrami算子$Δ$ G的一组n-1交换共形对称算子的度量的明确例子。本文解决了具有共形St{ä}ckel边界的紧致三维黎曼流形上的各向异性Calder{ó}n问题，即证明了这类流形的度量唯一地由在流形边界上测得的Dirichlet-to-Neumann映射确定，直到保持边界的dieomorphims.
英文摘要：
Conformally St{ä}ckel manifolds can be characterized as the class of n-dimensional pseudo-Riemannian manifolds (M, G) on which the Hamilton-Jacobi equation G($\nabla$u, $\nabla$u) = 0 for null geodesics and the Laplace equation --$Δ$ G $ψ$ = 0 are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples of metrics admitting a set of n--1 commuting conformal symmetry operators for the Laplace-Beltrami operator $Δ$ G. In this paper, we solve the anisotropic Calder{ó}n problem on compact 3-dimensional Riemannian manifolds with boundary which are conformally St{ä}ckel, that is we show that the metric of such manifolds is uniquely determined by the Dirichlet-to-Neumann map measured on the boundary of the manifold, up to dieomorphims that preserve the boundary.
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