Volume singular integro-differential equations in the electromagnetic diffraction problem

The problem of diffraction of electromagnetic waves on a 3-dimensional inhomogeneous dielectric body in a free space is considered. The original boundary value problem is reduced to a volume singular integro-differential equation. It is proved that for smooth data the solution from L₂ will necessary be continuous down to the boundary of the body. The equivalency between the boundary value problem and the integro-differential equation is proved. In addition, using pseudodifferential operators calculus, ellipticity of the operator of the problem is proved.