Maxwell’s equations

Abstract

Maxwell’s equations form an especially interesting example of the description of wave propagation phenomena. Thus we treat them in some detail. We start by formulating Maxwell’s equations, proving existence and uniqueness, and in section 8.3 treat the free space problem. Sections 8.1–8.3 are similar to sections 7.1–7.3; again the dimension of the null space of the underlying operator is infinite and the system is not elliptic. A consequence is the impossibility of generally estimating the first derivatives of the solutions up to the boundary. Thus we cannot use Rellich’s theorem to show compactness and therefore prove a selection theorem in section 8.5 after having discussed solutions in bounded domains in section 8.4. Finally, in section 8.6, we briefly treat exterior boundary value problems. Exterior boundary value problems and the existence of wave operators will be dealt with in Chapter 9 in more detail, where we shall present a unified approach to both Maxwell’s equations and the linearized system of acoustics.