AI 中文总结
本文针对可变声速下的波动方程逆问题，通过测量有限平面波及其互补解的边界数据，改进了早期刚性结果，利用卡尔曼估计提取未知系数，实现了二阶双曲算子最高阶项未知系数的唯一恢复。
AI 中文摘要
本文研究存在可变声速时波动方程的适定逆问题。我们证明，通过测量有限个平面波及其互补解的边界数据，可唯一恢复含时不变系数的二阶双曲算子最高阶项的未知系数。这改进了文献[10]、[11]中的早期刚性结果，这些结果比较的是黎曼度量生成的波动算子与欧氏度量生成的波动算子。我们比较两个具有相同低阶项、含时不变系数的一般二阶双曲算子，但要求其中一个算子关联的几何满足伪凸性条件、无焦散条件和张成条件。特别地，其中一个算子可以是声速接近常数的波动算子，另一个算子可以是任意的。为证明该结果，我们引入由入射平面波生成的广义平面波解的互补解概念，该互补解可在界面处光滑延拓广义平面波；未知系数出现在界面处的输运方程中，我们通过波动算子的一系列卡尔曼估计，可从该输运方程和边界数据中提取内部的未知系数。
英文摘要
In this article, we study formally determined inverse problems for wave equations in the presence of a variable sound speed. We prove that by measuring the boundary data of finitely many plane waves and their complementary solutions, one can uniquely recover the unknown coefficients of the highest order terms of a second order hyperbolic operator with time independent coefficients. This improves earlier rigidity results in [10], [11] which compared the wave operator generated by a Riemannian metric with the wave operator generated by the Euclidean metric. We compare two general second order hyperbolic operators with time independent coefficients, with the same lower order terms. However, we require the geometry associated with one of the operators to satisfy a pseudoconvexity condition, a no-caustics condition, and a spanning condition. In particular one of the operators could be a wave operator with the sound speed close to a constant and the other operator could be arbitrary. To prove the results, we introduce the notion of a complementary solution for a generalized plane wave solution generated by an incoming plane wave. The complementary solution extends smoothly, across an interface, the generalized plane wave. The unknown coefficients appear in a transport equation at the interface. We show that the unknown coefficients in the interior can be extracted from this transport equation, from the boundary data, via a sequence of Carleman estimates for the wave operator.
