Stability for the inverse source problems in elastic and electromagnetic waves

Abstract

This paper concerns the inverse source problems for the time-harmonic elastic and electromagnetic wave equations. The goal is to determine the external force and the electric current density from boundary measurements of the radiated wave field, respectively. The problems are challenging due to the ill-posedness and complex model systems. Uniqueness and stability are established for both of the inverse source problems. Based on either continuous or discrete multi-frequency data, a unified increasing stability theory is developed. The stability estimates consist of two parts: the Lipschitz type data discrepancy and the high frequency tail of the source functions. As the upper bound of frequencies increases, the latter decreases and thus becomes negligible. The increasing stability results reveal that ill-posedness of the inverse problems can be overcome by using multi-frequency data. The method is based on integral equations and analytical continuation, and requires the Dirichlet data only. The analysis employs asymptotic expansions of Green's tensors and the transparent boundary conditions by using the Dirichlet-to-Neumann maps. In addition, for the first time, the stability is established on the inverse source problems for both the Navier and Maxwell equations.