AI 中文总结
研究从单个边界测量对标量波动方程初始源和声速同时恢复，通过本构关系耦合未知量，利用微局部和卡尔曼框架，在特定几何条件下建立全局唯一性和利普希茨稳定性，还获部分数据结果。
AI 中文摘要
我们研究从单个边界测量中对标量波动方程的初始源和声速进行同时恢复。尽管在合适的几何假设下单独恢复任一参数已被充分理解，但同时恢复一般仍未解决，并且线性化问题的固有不稳定性进一步阻碍了稳定恢复。我们表明，当两个未知量通过由共同基础材料产生的规定本构关系耦合时，可以获得唯一性和稳定性。在由校准材料状态激发的定量非退化条件下，耦合反问题简化为单个反源问题。将Stefanov和Uhlmann的微局部和卡尔曼框架应用于此简化问题，我们在涉及严格凸叶状结构和测地线可见性的几何条件下建立了全局唯一性和利普希茨稳定性。我们还获得了部分数据结果，证明了在本构非退化条件成立的紧致可见区域中的局部唯一性和利普希茨稳定性。
英文摘要
We study the simultaneous recovery of the initial source and sound speed for the scalar wave equation from a single boundary measurement. Although the recovery of either parameter separately is well understood under suitable geometric hypotheses, simultaneous recovery remains open in general, and stable recovery is further obstructed by the inherent instability of the linearized problem. We show that both uniqueness and stability can be obtained when the two unknowns are coupled through a prescribed constitutive relation arising from a common underlying material. Under a quantitative nondegeneracy condition, motivated by calibrated material regimes, the coupled inverse problem reduces to a single inverse source problem. Applying the microlocal and Carleman framework of Stefanov and Uhlmann to this reduced problem, we establish global uniqueness and Lipschitz stability under geometric conditions involving strictly convex foliations and geodesic visibility. We also obtain partial-data results, proving local uniqueness and Lipschitz stability in compact visible regions where the constitutive nondegeneracy condition holds.
