发表机构
National Yang Ming Chiao Tung University; University of Duisburg-Essen（国立阳明交通大学; 杜伊斯堡-埃森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文证明三维及以上区域中各向异性Calderón问题在欧几里得度量附近的刚性：若光滑度量充分接近欧几里得度量且具有相同Dirichlet-to-Neumann映射，则必相差一个固定边界的微分同胚，方法结合调和坐标与Carleman估计。
AI 中文摘要
我们证明了在三维及更高维光滑有界连通区域上，各向异性Calderón问题在欧几里得度量附近的刚性。任何在适当Hölder范数下充分接近欧几里得度量的光滑黎曼度量，若具有相同的Dirichlet-to-Neumann映射，则必与欧几里得度量相差一个逐点固定边界的微分同胚。该结果适用于一般光滑各向异性扰动，无需解析或拟解析正则性假设、无需指定共形类、无需横向乘积结构，也不要求边界凸性。证明将调和坐标与针对紧支撑修正子的Carleman估计相结合，在由扰动决定的频率范围内获得二次Fourier估计，再通过频率分解得出刚性结论。
英文摘要
We prove rigidity near the Euclidean metric for the anisotropic Calderón problem on smooth bounded connected domains in dimensions three and higher. Any smooth Riemannian metric sufficiently close to the Euclidean metric in a suitable Hölder norm and having the same Dirichlet-to-Neumann map agrees with it up to a diffeomorphism fixing the boundary pointwise. The result applies to general smooth anisotropic perturbations, without analytic or quasianalytic regularity assumptions, a prescribed conformal class, or a transversal product structure, and requires no convexity of the boundary. The proof combines harmonic coordinates with a Carleman estimate for compactly supported correctors to obtain a quadratic Fourier estimate on a frequency range determined by the perturbation. A frequency decomposition then yields rigidity.
Comments24 pages
