In this work, we investigate the shape identification and coefficient determination associated with two time-dependent partial differential equations in two dimensions. We consider the inverse problems of determining a convex polygonal obstacle and the coefficient appearing in the wave and Schrodinger equations from a single dynamical data along with the time. With the far field data, we first prove that the sound speed of the wave equation together with its contrast support of convex-polygon type can be uniquely determined, then establish a uniqueness result for recovering an electric potential as well as its support appearing in the Schrodinger equation. As a consequence of these results, we demonstrate a uniqueness result for recovering the refractive index of a medium from a single far field pattern at a fixed frequency in the time-harmonic regime.

中文翻译：

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具有单一动态数据的瞬态逆问题的唯一性

在这项工作中，我们研究了与二维时间相关的两个偏微分方程相关的形状识别和系数确定。我们考虑了确定凸多边形障碍物的逆问题，以及从单个动力学数据以及时间出现在波和薛定谔方程中的系数。利用远场数据，我们首先证明波动方程的声速及其凸多边形类型的对比度支持可以唯一确定，然后建立恢复电位的唯一性结果及其出现在薛定谔方程。由于这些结果，