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

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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 Maxwell**Aufwärts:**Maxwell
equations (ME)**Vorherige Seite:**ME
and Dirac equation
collected and worked out by Wolfgang Sprößig, TU-Bergakademie
Freiberg