A trace theorem for solutions of the wave equation, and the remote determination of acoustic sources

The determination of sources of acoustic wave motion in several dimensions from remote measurements is of considerable interest in many applications, and the underlying mathematical problem is quite ill‐posed. We separate the source determination problem into a control problem for the wave equation and an inverse mixed initial‐boundary value problem, and concentrate on the latter, in which the initial data for a solution of the wave equation are to be determined from its trace on a time‐like hyperplane. Though the geometry of this problem is simple, it exhibits some of the central analytic difficulties of more complex problems. We prove a uniqueness theorem, give examples of instability, establish regularity properties of the trace, and locate noncompact classes of stable functionals. The existence of these noncompact classes shows that the problem is “partially well‐posed”, i.e. that smoothing in all directions is not required to regularize the problem, and distinguishes it from most other ill‐posed problems, such as backwards diffusion and analytic continuation.