Vector potential theory on nonsmooth domains in \(\mathbb{R}^ 3\) and applications to electromagnetic scattering
type: Item
instance of: scholarly article
title: Vector potential theory on nonsmooth domains in \(\mathbb{R}^ 3\) and applications to electromagnetic scattering
zbMATH Open document ID: 0877.35124
DOI: 10.1007/BF02649132
author: Dorina Mitrea
author: Marius Mitrea
author: Jill Pipher
published in: The Journal of Fourier Analysis and Applications
publication date: 1997-12-08T00:00:00Z
full work available at URL: https://eudml.org/doc/59505
review text: 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 …
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 31B10
Mathematics Subject Classification ID: 78A25
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 31B20
Mathematics Subject Classification ID: 78A30
Mathematics Subject Classification ID: 35-02
zbMATH DE Number: 1019261
zbMATH Keywords: Maxwell equations
zbMATH Keywords: Lipschitz domains
zbMATH Keywords: vector layer potentials
zbMATH Keywords: scatterings
reviewed by: Vasile Ionescu
MaRDI profile type: Publication
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