Regularity of weak solution to Maxwell's equations and applications to microwave heating

Abstract

In this paper we first study the regularity of weak solution for time-harmonic Maxwell's equations in a bounded anisotropic medium Ω. It is shown that the weak solution to the linear degenerate system, ∇×(γ(x) ∇× E )+ξ(x) E = J (x), x∈Ω⊂R 3 , is Hölder continuous under the minimum regularity assumptions on the complex coefficients γ( x) and ξ( x). We then study a coupled system modeling a microwave heating process. The dynamic interaction between electric and temperature fields is governed by Maxwell's equations coupled with an equation of heat conduction. The electric permittivity, electric conductivity and magnetic permeability are assumed to be dependent of temperature. It is shown that under certain conditions the coupled system has a weak solution. Moreover, regularity of weak solution is studied. Finally, existence of a global classical solution is established for a special case where the electric wave is assumed to be propagating in one direction.