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

TL;DR

This paper reviews boundary value and scattering problems for bodies in physics, reformulating them within the abstract theory of composites to analyze responses like dielectric polarizability and electromagnetic scattering.

Contribution

It unifies the treatment of boundary value and scattering problems using the abstract theory of composites, extending previous work to include scattering responses.

Findings

Scattering responses can be computed from integrals over inclusions.

Boundary value problems are reformulated as problems in the abstract theory of composites.

Effective operators like the Dirichlet-to-Neumann map are used to analyze responses.

Abstract

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.