Inverse elastic obstacle scattering problems by monotonicity method

We consider the elastic wave scattering problem involving rigid obstacles. This work addresses the inverse problem of reconstructing the position and shape of such obstacles using far-field measurements. A novel monotonicity-based approach is developed for this purpose. By factorizing the far-field operator and utilizing the existence of localized wave functions, we derive a shape characterization criterion for the obstacle boundary. The proposed method employs monotonicity tests to determine the geometric relationship between any given test domain and the actual scatterer. As a result, the shape and location of rigid elastic obstacles can be uniquely identified without requiring any initial guesses or prior knowledge of the physical parameters of the homogeneous background medium.