The integral equation method in electromagnetic scattering (Q1071171)

The problem of scattering of an electromagnetic time harmonic wave by a bounded perfectly conducting object with a Lyapunov boundary S is considered. The problem is formulated in an appropriate Banach space as the integral equation \((1)\quad {\mathfrak a}-R_ k {\mathfrak a}={\mathfrak c}\) for the surface current \({\mathfrak a}=\hat {\mathfrak n}\times {\mathfrak H}\), where \(R_ k {\mathfrak a}=\hat {\mathfrak n}\times [\nabla \times (1/2\pi)\int_{S}(e^{ikR}/R){\mathfrak a} d\quad s].\) Here \(\hat{\mathfrak n}\) denotes a unit normal vector to S, R is a distance between the observation and integration points, k is a wavenumber and \({\mathfrak c}=2\hat {\mathfrak n}\times {\mathfrak H}^{inc}\). \({\mathfrak H}\) and \({\mathfrak H}^{inc}\) are total and incident magnetic fields, respectively. Starting from the case \(k=0\) the authors establish basic properties of the operator \(R_ 0\) including existence, Hölder continuity and compactness. They prove that the spectral radius of \(R_ 0\) is strictly less than one and hence the Neumann series for \((I-\lambda R_ 0)^{- 1}\) converges for \(| \lambda | \leq 1.\) Proceeding to \(k>0\) the authors show that for sufficiently small k the Neumann series for \((I-R_ k)^{-1}\) also converges. This enables them to propose a successive approximation method of solving the integral equation (1), based on the iteration: \({\mathfrak a}^{(n+1)}=R_ k a^{(n)}+{\mathfrak c}.\) Finally, the authors establish an equivalence between the solution of the integral equation and the solution of the boundary value problem.