This paper is concerned with the inverse time harmonic elastic scattering of multiple small and well-resolved cavities in two dimensions. We extend the so-called DORT method to the inverse elastic scattering so that selective focusing can be achieved on each cavity with far field measurements. A rigorous mathematical justification that relates the corresponding eigenfunctions of the time reversal operator to the locations of cavities is presented based on the asymptotic analysis of the far field operator and decaying property of oscillatory integrals. We show that in the regime $a\ll k^{-1}\ll L$, where $a$ denotes the size of cavity, $k$ is the compressional wavenumber $\kp$ or shear wavenumber $\ks$, and $L$ is the minimal distance between the cavities, each cavity gives rise to five significant eigenvalues and the corresponding eigenfunction generates an incident wave focusing selectively on that cavity. Numerical experiments are given to verify the theoretical result.

翻译：本文研究二维空间中多个微小且可分辨空腔的逆时谐弹性散射问题。我们将DORT方法推广至逆弹性散射，从而可利用远场测量实现对每个空腔的选择性聚焦。基于远场算子的渐近分析和振荡积分的衰减特性，给出了时间反演算子对应的本征函数与空腔位置之间关系的严格数学证明。我们证明，在$a\ll k^{-1}\ll L$的尺度关系下（其中$a$表示空腔尺寸，$k$为压缩波数$\kp$或剪切波数$\ks$，$L$为空腔间最小距离），每个空腔会产生五个显著特征值，且对应的本征函数可生成选择性聚焦于该空腔的入射波。数值实验验证了该理论结果。