Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation
Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation
复制标题
黎曼波动方程的狄利克雷到诺依曼算子的共域刚性
DOI：
10.1090/tran/7630
复制
发表时间：
2019
影响因子：
1.3
通讯作者：
A. Mansouri
中科院分区：
文献类型：
--
作者：
Tristan Milne;A. Mansouri
研究了紧致黎曼流形上黎曼波动方程的Dirichlet to Neumann算子。如果黎曼流形被建模为弹性介质，则该算子表示当流形通过边界振动而运动时流形边界上的观察者可用的数据。我们研究了狄利克雷诺依曼算子时，施加振动和数据记录在不相交的集，一个有用的设置的应用程序。我们证明，该操作员确定狄利克雷诺依曼操作源和观察是在同一个集合上，提供了一个频谱条件的拉普拉斯-贝尔特拉米算子的流形是满意的。我们证明了这一点，提供了一个可实施的程序，用于确定一部分的黎曼流形附近的地区，其中应用源。利用已有的结果，一个直接的推论是，一个紧凑的黎曼流形可以重建从狄利克雷诺依曼算子的来源和意见是不相交的集。
We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.