Stability estimates for the X-ray transform of tensor fields and boundary rigidity
Stability estimates for the X-ray transform of tensor fields and boundary rigidity
复制标题
张量场 X 射线变换和边界刚度的稳定性估计
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
G. Uhlmann
中科院分区：
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作者：
Plamen Stefanov;G. Uhlmann
我们研究 Rn 中域的边界刚性问题：是由距离函数 g.x 唯一确定的黎曼度量，直到固定边界的微分同胚作用； y/ 已知所有边界点 x andy？米歇尔推测这对于简单的指标来说是正确的。在本文中，我们首先研究线性化问题，其中包括从测地线 X 射线积分变换 Ig 确定对称 2-张量，直至潜在项。我们证明了正规算子 Ng D I g Ig 是一个伪微分算子，前提是 g 很简单，找到它的主符号，识别它的核，并构造一个微局部参数矩阵。我们证明了与线性问题相关的亚椭圆型稳定性估计。接下来，我们应用此估计来表明，对于给定的简单度量 g，直到潜在项，线性问题的唯一可解性意味着该 g 附近的非线性边界刚性问题的局部唯一性。
We study the boundary rigidity problem for domains in Rn: is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function g.x; y/ known for all boundary points x andy? It was conjectured by Michel that this was true for simple metrics. In this paper, we study the linearized problem first which consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transformIg . We prove that the normal operator Ng D I g Ig is a pseudodifferential operator provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove hypoelliptic type of stability estimate related to the linear problem. Next we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the non-linear boundary rigidity problem near that g.