  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.

Analysis of PDEs