Vector potential theory on nonsmooth domains in \(\mathbb{R}^ 3\) and applications to electromagnetic scattering
The paper is an extensive study of the problems arising in the analysis of electromagnetic wave propagation in a homogeneous, isotropic medium that occupies the exterior of a bounded domain \(\Omega\subset\mathbb{R}^3\), with the electric conductivity \(\sigma\geq 0\), the electric permitivity \(\varepsilon>0\), and magnetic permeability \(\mu\). The electric and magnetic fields \(E\) and \(H\) satisfy the Maxwell equations: \[ \text{curl }E(X,t)+ \mu(\partial H(X,t)/\partial t)=0\quad\text{in }(\mathbb{R}^3\backslash\overline\Omega)\times\mathbb{R}, \] \[ \text{curl }H(X,t)-\varepsilon(\partial E(X,t)/\partial t)= J(X,t)\quad\text{in }(\mathbb{R}^3\times\overline\Omega)\times\mathbb{R}. \] \(J\) is the current density. Although the authors do not consider this system of equations in all its generality, it remains a very difficult problem to study and solve, as it is known from the vast literature in the domain. They, themselves, have their substantial contribution to the development of the literature in the last five years. Surveying the principal results obtained in the field in the last half a century, the authors present their own point of view in the approach of the problem: They extend the vector potential theory to the electromagnetic scattering. Single-layer, double layer, and Newtonian potentials, known from many other boundary problems in mathematical physics, appear in their considerations. The authors establish, generally, principal properties of these potentials as: behaviour in the existence domain, continuity in passing the boundary, behaviour of normal derivatives. All these properties are expressed in terms of: the space of the solutions, the nature of the boundaries, and the class of the given data. Many new properties with respect to the boundary integrals and boundary derivative operators appear in the analyzed cases, and new notions are introduced for synthetical reasons, as the divergence surface. One of the important parts of the paper is the sixth one, devoted to the \(L^p\)-theory for Maxwell equations and related system, where many important theorems concerning existence, uniqueness, and properties of the solutions are proved. Special attention is payed by the authors to the case \(p=1\), called by them ``Atomic theory for the Maxwell equations'', where, also, many results are gathered in a special chapter. The last parts of the paper are devoted to: \(L^p\) theory for the nonregularity boundary value problems, non-homogeneous boundary value problems for the decoupled Maxwell equations and to the inverse electromagnetic scattering problem. The last chapter, very short but very important in our opinion, contains some open problems and questions, where unsolved problems, possibilities of their solution and possible refinements of the already obtained results are presented. It could be considered as a promise of the authors that they intend to continue the study of problems to which they have presented remarkable contributions.
scientific article; zbMATH DE number 1019261
scientific article