各向异性Calderón问题中的拟解析性与几何刚性
Quasianalyticity and geometric rigidity in anisotropic Calderón's problem
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中文总结 AI 辅助
本文针对n≥3维未解决的各向异性Calderón问题，在两种互补情形下建立唯一性结果，揭示了正则性、几何结构与边界访问间的权衡关系。
中文摘要 AI 辅助
n≥3维的各向异性Calderón问题对于一般光滑度量仍未解决。我们在两个互补的情形中建立了唯一性结果：第一种情形下，拟解析函数的恒等原理传播边界信息，在包括部分边界情形在内的一般几何中得到唯一性；在规定的法几何下，仅需在特定方向上满足拟解析性。第二种情形下，合适的对称性或单侧序假设可在C^∞正则性下，通过全部或受限边界访问得到唯一性。综上，结果展现了正则性、几何结构与边界访问间的权衡：拟解析性在一般几何中提供延拓，而对称性或单侧序在C^∞正则性下替代该延拓。
英文摘要
The anisotropic Calderón problem of determining a smooth Riemannian metric from boundary measurements, up to a boundary-fixing diffeomorphism, remains open in dimensions $n\ge3$~\cite{uhlmann2009electrical}. We establish unique identifiability results in two complementary regimes. In the first, the identity principle for quasianalytic functions propagates boundary information and yields unique identifiability on compact manifolds without a prescribed product structure, including a partial-data consequence; under a prescribed normal geometry, quasianalyticity is needed only in the distinguished direction. In the second, suitable symmetry or one-sided ordering assumptions lead to unique identifiability at $C^\infty$ regularity with full or restricted boundary access. Taken together, the results exhibit a tradeoff among regularity, geometric structure, and boundary access: quasianalyticity supplies continuation when no global product structure is prescribed, while symmetry or one-sided order replaces that continuation at $C^\infty$ regularity.
发表机构
- Key Laboratory of Algebraic Lie Theory and Analysis, Ministry of Education, School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院代数李理论与分析教育部重点实验室）
- Department of Mathematics, City University of Hong Kong（香港城市大学数学系）
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