Silent sources on a surface for the Helmholtz equation and decomposition of L² vector fields

Overview & Synopsis

We study an inverse source problem with right hand side in divergence form for the Helmholtz equation, whose underlying model can be related to weak scattering from thin interfaces. This inverse problem is not uniquely solvable, as the forward operator has infinite-dimensional kernel. We present a decomposition of (not necessarily tangent) vector fields of L 2-class on a closed Lipschitz surface in R 3 , which allows one to discuss an ansatz for the solution and constraints that restore uniqueness. This work can be seen as a generalization of references [4, 6] dealing with the Laplace equation, but in the Helmholtz case new ties arise between the observations from each side of the surface. Our proof is based on properties of the Calderón projector on the boundary of Lipschitz domains, that we establish in a H-1 × L 2 setting. [4] L. Baratchart, C. Gerhards, and A. Kegeles. Decomposition of l 2-vector fields on lipschitz surfaces: characterization via null-spaces of the scalar potential. SIAM Journal on Mathematical Analysis, 2021. [6] L. Baratchart, C. Villalobos Guillén, D. P. Hardin, M. C. Northington, and E. B. Saff. Inverse potential problems for divergence of measures with total variation regularization. Foundations of Computational Mathematics, Nov 2019.