Silent Sources on a Surface for the Helmholtz Equation and Decomposition of \({L^2}\) Vector Fields

We present a decomposition of R 3 -valued vector fields of L 2 -class on the boundary of a Lipschiz domain in R 3 , that relates to an inverse source problem with source term in divergence form for the Helmholtz equation.Applications thereof include weak scattering from thin interfaces.The inverse problem is not uniquely solvable, as the forward operator has infinite-dimensional kernel.The proposed decomposition brings out constraints that can be used to restore uniqueness.This work subsumes the one in [3] dealing with the Laplace equation, and shines a light on new ties that arise in the Helmholtz case between solutions on each side of the surface.Our approach uses properties of the Calderón projectors on the boundary of Lipschitz domains, that we carry out in the L 2 × H -1 setting to handle L 2 vector fields in the style of [29], dwelling on results from [18]; it applies for any complex wave number.