Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens

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.

Keywords:

• 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

AMS Subject Classifications:

• 31B10
• 35A02
• 35Q61
• 35S15
• 78A45

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Russian Science Foundation under [grant number 14-11-00344].