Existence and Regularity of the solutions to Maxwell’s Equations
Dr. Gaofeng Zheng
Department of Mathematics
University of Central Florida
Abstract: In this talk, we discuss some recent results on Maxwell's equations. The first part is concerned with the mathematical analysis of the electromagnetic wave scattering by anunbounded dielectric medium, which is mounted on a perfectly conducting infnite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwells equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax-Milgram theorem.
In the second part, we derive local estimates of singular positive solutions to a system of nonlinear Maxwell equations in the plane R2, based on which a classification theorem of general positive solutions is established. The refined singularity of general positive solutions is also investigated by employing the theory of infinite dimensional dynamical systems.
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