A note on div-curl lemmas for Maxwell interface problems

Combining knowledge of the curl and divergence of a vector field $\mathbf{E}$ to obtain information about its spatial regularity has proven to be a very useful technique in the treatment of the Maxwell equations. We consider the interface problem, where the permittivity $\varepsilon$ is discontinuous across a surface. An important theorem by Weber can be generalized to allow for a jump of $\varepsilon\mathbf{E}$ in normal direction across the interface. We use a Helmholtz decomposition to deduce this from the Weber result by a reduction to the task of proving higher regularity for solutions of elliptic transmission problems. For electric boundary conditions, the result was shown recently by Dohnal, Ionescu-Tira and Waurick. We extend the result to the magnetic case using similar arguments. The main goal of this note is to present the proofs in detail. In particular, we keep track of how the constants depend on the permittivity. This information is useful for approximation arguments.