波速与初始源同时恢复中的Lipschitz稳定性
Lipschitz stability in the simultaneous recovery of wave speed and initial source
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中文总结 AI 辅助
针对双曲方程反问题，提出同时恢复波速系数与初始位移的方法，利用变换与求导得到耦合系统，结合加权能量与Carleman估计，在Sobolev范数下建立Lipschitz稳定性，将系数辨识推广至联合辨识。
中文摘要 AI 辅助
我们考虑一个双曲方程的反问题。目标是从一对Dirichlet-Neumann Cauchy数据中同时恢复空间变化系数（其平方根代表波速）和初始位移。通过引入适当的变换并对所得方程关于时间求导，我们推导出未知系数与初始位移之差的耦合系统。将加权能量估计与Carleman估计相结合，我们在适当的Sobolev范数下建立了两个未知量的Lipschitz稳定性估计。证明依赖于先验正则性假设、初始数据的非退化条件以及控制系数与初始位移差之间相互作用的结构不等式。我们的结果将仅系数的稳定性结果推广到系数与初始位移同时辨识问题。
英文摘要
We consider a inverse problem for a hyperbolic equation. The objective is to simultaneous recover both the spatially varying coefficient, whose square root represents the wave speed, and the initial displacement from a single pair of Dirichlet-Neumann Cauchy data. By introducing a suitable transformation and differentiating the resulting equation in time, we derive a coupled system for the differences of the unknown coefficient and initial displacement. Combining weighted energy estimates with Carleman estimates, we establish a Lipschitz stability estimate for both unknown quantities in appropriate Sobolev norms. The proof relies on a priori regularity assumptions, non-degeneracy conditions on the initial data, and structural inequalities controlling the interaction between the coefficient and initial-displacement differences. Our result extends coefficient-only stability results to a simultaneous coefficient--initial-displacement identification problem.
发表机构
- School of Mathematics, Jilin University（吉林大学数学学院）
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