The method of layer potentials in electromagnetic scattering theory on nonsmooth domains
The author reconsiders the classical interior as well as the exterior boundary-value problems connected with the time-harmonic electromagnetic fields and gives some sufficient conditions which guarantee the existence and uniqueness of the solution when the main domain consists of an arbitrary Lipschitz domain \(\Omega\) in \(\mathbb{R}^3\) while the boundary data are in \(L^p(\partial\Omega)\) with \(2- \varepsilon\leq p\leq 2+ \varepsilon\). Here \(\varepsilon> 0\) stands for a certain small number depending only on the wave number as well as on the Lipschitz character of \(\Omega\). By Lipschitz domain one means a bounded, open and simply connected domain with connected boundary \(\partial\Omega\) such that \(\partial\Omega\) can be covered by a finite family of open cylinders having some particular properties. The solutions are expressed in terms of classical layer-potential operators which reduce the problems to weakly singular integral equations. The expansions of the solutions in series of electric dipoles are also studied. The problems are formulated separately for both the scattered electric and the scattered magnetic field vectors. The results are expressed in the form of eight theorems which are based on eleven lemmas. Notice that all the boundary conditions considered by the author are of Dirichlet type (not impedance or transmitting type, for example).
scientific article; zbMATH DE number 755406
scientific article