A quasilinear transmission problem with application to Maxwell equations with a divergence-free D-field

Maxwell equations in the absence of free charges require initial data with a divergence-free displacement field D. In materials in which the dependence D = D(E) is nonlinear the quasilinear problem (SIC).D(epsilon) = 0 is hence to be solved. In many applications, e.g. in the modelling of wave packets, an approximative asymptotic ansatz of the electric field E is used, which satisfies this divergence condition at t = 0 only up to a small residual. We search then for a small correction of the ansatz to enforce SIC).D(epsilon) = 0 at t = 0 and choose this correction in the form of a gradient field. In the usual case of a power type nonlinearity in D(epsilon) this leads to the sum of the Laplace and p-Laplace operators. We also allow for the medium to consist of two different materials so that a transmission problem across an interface is produced. We prove the existence of the correction term for a general class of nonlinearities and provide regularity estimates for its derivatives, independent of the L-2-norm of the original ansatz. In this way, when applied to the wave packet setting, the correction term is indeed asymptotically smaller than the original ansatz. We also provide numerical experiments to support our analysis. (C)& nbsp;2022 The Author(s). Published by Elsevier Inc.

Automated summary

This paper investigates the requirement of initial data with a divergence-free displacement field D in the absence of free charges in Maxwell equations. In the case of nonlinear dependence D = D(E), the quasilinear problem (SIC).D(epsilon) = 0 needs to be solved. The paper introduces an approximate asymptotic ansatz of the electric field E and a small correction term to satisfy the initial condition. The existence of the correction term and regularity estimates for its derivatives are proven, and numerical experiments are provided to support the analysis.