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Maxwell equations in the absence of free charges require initial data with adivergence free displacement field $\mathcal D$. In materials in which thedependence $\mathcal D=\mathcal D(\mathcal E)$ is nonlinear the quasilinearproblem $\nabla\cdot\mathcal D(\mathcal E)=0$ is hence to be solved. In manyapplications, e.g. in the modelling of wave-packets, an approximativeasymptotic ansatz of the electric field $\mathcal E$ is used, which satisfiesthis divergence condition at $t=0$ only up to a small residual. We search thenfor a small correction of the ansatz to enforce $\nabla\cdot\mathcal D(\mathcalE)=0$ at $t=0$ and choose this correction in the form of a gradient field. Inthe usual case of a power type nonlinearity in $\mathcal D(\mathcal E)$ thisleads to the sum of the Laplace and $p$-Laplace operators. We also allow forthe medium to consist of two different materials so that a transmission problemacross an interface is produced. We prove the existence of the correction termfor a general class of nonlinearities and provide regularity estimates for itsderivatives, independent of the $L^2$-norm of the original ansatz. In this way,when applied to the wave-packet setting, the correction term is indeedasymptotically smaller than the original ansatz. We also provide numericalexperiments to support our analysis.

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