Title: Stable determination by a single measurement, scattering bound and regularity of transmission eigenfunction Show Chinese title

Title: 单次测量的稳定确定性，散射界和透射本征函数的正则性

Abstract: In this paper, we study an inverse problem of determining the cross section of an infinitely long cylindrical-like material structure from the transverse electromagnetic scattering measurement. We establish a sharp logarithmic stability result in determining a polygonal scatterer by a single far-field measurement. The argument in establishing the stability result is localized around a corner and can be as well used to produce two highly intriguing implications for invisibility and transmission resonance in the wave scattering theory. In fact, we show that if a generic medium scatterer possesses an admissible corner on its support, then there exists a positive lower bound of the $L^2$-norm of the associated far-field pattern. For the transmission resonance, we discover a quantitative connection between the regularity of the transmission eigenfunction at a corner and its analytic or Fourier extension. Show Chinese abstract

Abstract: 在本文中，我们研究了一个反问题，即从横向电磁散射测量中确定无限长圆柱形材料结构的横截面。 我们建立了一个在单个远场测量下确定多边形散射体的尖锐对数稳定性结果。 在建立稳定性结果的过程中，论证集中在角落附近，也可以用来产生关于波散射理论中隐身和传输共振的两个高度有趣的推论。 事实上，我们证明，如果一个一般的介质散射体在其支撑上具有可接受的角落，那么相关远场模式的$L^2$-范数存在正的下界。 对于传输共振，我们发现角落处传输本征函数的正则性与其解析或傅里叶延拓之间存在定量联系。