Nächste Seite:Picard' s extended MaxwellAufwärts:Maxwell equations (ME)Vorherige Seite:ME and Dirac equation

### Non-linear Maxwell equations

We consider the following well-posed initial-boundary value problem:

Here denote  a bounded domain in . The electric and magnetic induction are given in such a way that

Introducing the charge density  by the relation  we obtain
Furthermore we have
In case of no current in the space we find  and so  From the above equations () we get
Assume the material relations  and . We consider the linearized system

Here  and  are matrices which are assumed to be symmetric. Compatibility equations follows easily:

where we used that . System () is versuitable for a proof of existence and uniqueness if  and  but not in the nonlinear case. Therefore consideration of the system
Finally the problem will be reduced to the symmetric hyperbolic system
where  and  are  symmetric matrices such that
where  is the fully antisymmetric Ricci tensor.

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.

Nächste Seite:Picard' s extended MaxwellAufwärts:Maxwell equations (ME)Vorherige Seite:ME and Dirac equation
collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg