Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens. Issue 8 (11th June 2017)

• Record Type: Journal Article
• Authors: Smirnov, Yu. G.; Tsupak, A. A.
• Is Part Of: Applicable analysis. Volume 96:Issue 8(2017)
• Page Start: 1326
• Page End: 1341
• Publication Date: 2017-06-11
• Subjects: Maxwell's equations -- diffraction problem -- inhomogeneous lossless bodies -- perfectly conducting screens -- boundary value problem -- singular integro-differential equation -- pseudodifferential operator -- Fredholm property -- existence and uniqueness of a solution; 31B10 -- 35A02 -- 35Q61 -- 35S15 -- 78A45; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2016.1188289
• Languages: English
• ISSNs: 0003-6811

Abstract

Abstract : New vector problem of electromagnetic wave diffraction by a system of non-intersecting three-dimensional inhomogeneous dielectric bodies and infinitely thin screens is considered in a quasiclassical formulation as well as the classical problem of diffraction by a lossless inhomogeneous body. In both cases, the original boundary value problem for Maxwell's equations is reduced to integro-differential equations in the regions occupied by the bodies (and on the screen surfaces). The integro-differential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Uniqueness of solutions is proved under the realistic hypothesis of discontinuity of the dielectric permittivity the boundary of a volume scatterer. This result allowed to establish invertibility of the integro-differential operator in sufficiently broad spaces. For the problem of diffraction on dielectrics and surface conductors, theorem on smoothness of a solution is proved under assumption of data smoothness. The latter implies equivalence between the differential and integral formulations of the scattering problem. The matrix integro-differential operator is proved to be a Fredholm invertible operator. Thus, the existence of a unique solution to both problems is established.