高频固定频率下黎曼度量的各向异性 Calderón 问题
The anisotropic Calderón problem for Riemannian metrics at high fixed frequency
- CNRS, Université Paris-Est Créteil, Université Gustave Eiffel（法国国家科学研究中心、巴黎东马恩拉瓦莱大学、古斯塔夫埃菲尔大学）
- Department of Mathematics, University of California, Irvine（加州大学欧文分校数学系）
- Department of Mathematics, Indian Institute of Technology Bombay（印度理工学院孟买分校数学系）
- Department of Mathematics, University of Washington（华盛顿大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文证明了在固定高频下，黎曼流形上 Helmholtz 方程的 Dirichlet-to-Neumann 映射相等可推出透镜数据相等，从而建立 Calderón 问题与透镜刚性之间的高频桥梁，并给出定量版本。
AI中文摘要：
我们研究光滑紧致、无捕获且具有严格凸边界的黎曼流形上 Helmholtz 方程的逆边值问题。我们证明，给定两个这样的度量，对于足够大但固定的频率 λ，它们的 Dirichlet-to-Neumann 映射相等意味着它们的透镜数据相等，直至一个平滑的固定边界的微分同胚。这建立了 Calderón 问题与透镜刚性之间的高频桥梁。我们还证明了该结果的定量版本，该版本在合适的黎曼度量有界集中一致有效。关键地，我们证明了集中在最大测地线上的高斯波束型解在边界附近分裂为出射/入射部分，使得出射部分的相位包含出射点、方向和传播时间的信息。透镜数据的恢复随后通过仔细的驻相分析和边界积分恒等式进行。
英文摘要：
We study an inverse boundary value problem for the Helmholtz equation on a smooth compact non-trapping Riemannian manifold with strictly convex boundary. We prove that, given two such metrics, for sufficiently large but fixed frequency $λ$, equality of their Dirichlet-to-Neumann maps implies equality of their lens data, up to a smooth boundary-fixing diffeomorphism. This establishes a high-frequency bridge between the Calderón problem and lens rigidity. We also prove a quantitative version of this result uniformly valid in a suitable bounded set of Riemannian metrics. Crucially, we show that Gaussian beam type solutions concentrating on maximal geodesics, split near the boundary into outgoing/incoming parts, such that the phase of the outgoing part contains information on exit point, direction, and travel time. The recovery of lens data then proceeds by a careful stationary phase analysis and a boundary integral identity.
