关于结构化弹性密度与初始状态的被动恢复
On passive recovery of structured elastic density and initial states
- Northeast Normal University（东北师范大学）
- City University of Hong Kong（香港城市大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文研究三维弹性波动方程中密度与初始状态的同时恢复，利用边界位移迹线数据，通过拉普拉斯展开和拉梅模式，在特定条件下证明了密度轮廓的唯一性。
中文摘要 AI 辅助
我们研究三维各向同性弹性波动方程中（可变）质量密度、初始位移和初始速度的同时恢复问题，其中拉梅参数为已知常数。数据是封闭边界上的完整位移迹线。我们首先假设密度加权的初始位移和速度在一个空间方向上具有固定的已知轮廓。零频拉普拉斯展开的$s^0$和$s^1$系数识别出这些加权状态。$s^2$和$s^3$系数则给出密度差的静态拉梅正交恒等式。通过$s^3$阶的精确差展开具有$O(s^4)$余项，在有界空间集以及有界密度对比和加权状态类上一致成立。在两个初始状态对齐且密度加权初始速度具有非零矩的条件下，两个密度恒等式简化为常向量静态变换。当轮廓族是双边拉普拉斯非退化时，两个相反的弹性零相位给出一个或两个固定垂直密度轮廓的唯一性。当$\lambda+\mu\ne0$时，每个非零正规根具有部分重数$2$和$1$，并允许长度为二的约当链。相关的多项式-指数拉梅模式产生双边轮廓变换的导数。由此得到的埃尔米特-拉普拉斯系统给出对齐类的唯一性，该类允许最多四个固定垂直轮廓和独立水平系数，前提是轮廓系统是埃尔米特-拉普拉斯非退化的。一个紧支撑轮廓的不同平移提供了一个显式的四轮廓类。我们还证明了对齐约化的刚性，并展示了在无限制密度上约化变换的无限维核。
英文摘要
We study simultaneous recovery of the (variable) mass density, initial displacement, and initial velocity for the three-dimensional isotropic elastic wave equation with known constant Lamé parameters. The data are the complete displacement trace on an enclosing boundary. We first assume that the density-weighted initial displacement and velocity have fixed known profiles in one spatial direction. The $s^0$ and $s^1$ coefficients of the zero-frequency Laplace expansion identify these weighted states. The $s^2$ and $s^3$ coefficients then give static Lamé orthogonality identities for the density difference. The exact difference expansion through order $s^3$ has an $O(s^4)$ remainder, uniformly on bounded spatial sets and bounded density-contrast and weighted-state classes. Under alignment of the two initial states and a nonzero moment of the density-weighted initial velocity, the two density identities reduce to a constant-vector static transform. Two opposite elastic null phases give uniqueness for one or two fixed vertical density profiles when the profile family is two-sided Laplace nondegenerate. When $λ+μ\ne0$, each nonzero normal root has partial multiplicities $2$ and $1$ and admits a length-two Jordan chain. The associated polynomial--exponential Lamé mode produces derivatives of the bilateral profile transforms. The resulting Hermite--Laplace system gives uniqueness for aligned classes with up to four fixed vertical profiles and independent horizontal coefficients, provided that the profile system is Hermite--Laplace nondegenerate. Distinct translations of one compactly supported profile provide an explicit four-profile class. We also prove rigidity of the alignment reduction and exhibit an infinite-dimensional kernel for the reduced transform on unrestricted densities.
