The vector Helmholtz equation revisited: Inverse obstacle scattering

Abstract

The vector Helmholtz equation, from a mathematical point of view, provides a generalization of the time-harmonic Maxwell equations for the propagation of time-harmonic electromagnetic waves. After reviewing some classic results on the two main exterior boundary value problems for the vector Helmholtz equation, i.e., the so-called electric boundary condition and the magnetic boundary condition, we prove reciprocity results for scattering of plane waves and point sources. Then we make use of them for obtaining uniqueness results for the inverse obstacle scattering problem to determine the shape of the scatterer from knowing the far field pattern of the scattered waves and results on the so-called far field operator for the two scattering problems. These results are generalizations of corresponding results for the Maxwell equations and analogues to results for the Dirichlet and Neumann boundary condition for the scalar Helmholtz equation. We also briefly consider the extension of the so-called DB boundary condition from the Maxwell equations to the vector Helmholtz equation.