标题： 通过单次远场测量稳定确定阻抗障碍物 显示英文标题

标题： Stable determination of an impedance obstacle by a single far-field measurement

摘要： 我们在$\mathbb{R}^2$中建立了对确定阻抗障碍物的对数类型的尖锐稳定性估计。 障碍物具有一般的多边形形状，阻抗参数可以是变化的。 我们通过使用一个远场模式建立了稳定性结果，这在反散射理论中是一个长期存在的问题。 这是文献中首次通过单个远场测量来确定阻抗障碍物的稳定性结果。 如果障碍物具有通常的多边形形状，则在确定障碍物的稳定性方面是通过修改后的豪斯多夫距离建立的，并且与边界阻抗参数无关。 如果障碍物进一步已知为凸的，则在同时确定障碍物和边界阻抗的稳定性方面是通过经典的豪斯多夫距离建立的。 在为建立上述稳定性结果而开发的数学策略中，有若干技术上的新思路和发展。 首先，稳定性分析是在一个角落点以微局部的方式进行的。 其次，我们的稳定性估计建立了障碍物的几何配置与角落点处波场消失阶数之间的显式关系。 第三，我们开发了新的误差传播技术，以处理角落处波场的奇异性以及处理阻抗边界条件。 显示英文摘要

摘要： We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in $\mathbb{R}^2$. The obstacle is of general polygonal shape and the impedance parameter can be variable. We establish the stability results by using a single far-field pattern, which constitutes a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. If the obstacle is of a generally polygonal shape, the stability in determining the obstacle is established in terms of a modified Hausdorff distance and is independent of the boundary impedance parameter. If the obstacle is further known to be convex, the stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. There are several technical novelties and development in the mathematical strategy developed for establishing the aforementioned stability results. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships among the geometric configurations of the obstacle and the vanishing order of the wave field at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner as well as to tackle the impedance boundary condition.