A unifying perspective on linear continuum equations prevalent in physics. Part VII: Boundary value and scattering problems

We consider simply connected bodies or regions of finite extent in space or space-time and write conservation laws associated with the equations in Parts I-IV. We review earlier work where, for elliptic equations,the boundary value problem is reformulated as a problem in the abstract theory of composites and the associated effective operator is equated with the Dirichlet-to-Neumann map that governs the response of the body. The dielectric polarizability problem and acoustic and electromagnetic scattering by an inclusion are formulated as problems in the extended abstract theory of composites. The scattering response can be determined from appropriate integrals over the inclusion.