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Picard' s extended Maxwell system

It is considered the following system:
\begin{eqnarray*}\phi_t + {\rm div} D&=& q,\\D_t+D_E\cdot\nabla\phi +{\rm rot......rm rot} E + B_H\cdot \nabla \psi &=&0,\\\psi_t+{\rm div} B&=&0\end{eqnarray*}
where $D_E\cdot\nabla=\sum_{i=1}^3D_{E_1}\partial _i$$B_H\cdot\nabla=\sum_{i=1}^3B_{H_1}\partial _i$$\phi,\psi$ are artificial functions with the initial conditions:
\begin{eqnarray*}\phi\vert _{t=0}=0,\quad \psi\vert _{t=0}=0\end{eqnarray*}
and the boundary conditions
\begin{eqnarray*}\phi\vert _{\partial G}=0,\quad\psi\vert _{\partial G}=0.\end{eqnarray*}
A late reference: R. H. Picard, W.M. Zajaczkowski (1995) Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations, MMAS 18, 169-199.

collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg