SILENT SOURCES ON A SURFACE FOR THE HELMHOLTZ EQUATION AND DECOMPOSITION OF L² VECTOR FIELDS

We present a decomposition of \BbbR³-valued vector fields of L²-class on the boundary of a Lipschitz domain in \BbbR³ that relates to an inverse source problem with a 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 an infinite-dimensional kernel. The proposed decomposition brings out constraints that can be used to restore uniqueness. This work subsumes the one in [L. Baratchart, C. Gerhards, and A. Kegeles, SIAM J. Math. Anal., 53 (2021), pp. 4096-4117] 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 Calder\' on projectors on the boundary of Lipschitz domains that we carry out in the L² \timesH^-1 setting to handle L² vector fields in the style of [R. Torres and G. Welland, Indiana Univ. Math. J., 42 (1993), pp. 1457-1485], dwelling on results from [S. Hofmann, M. Mitrea, and M. Taylor, Int. Math. Res. Not., IMRN 2010 (2010)]; it applies for any complex wave number.