Spaces
In the following, we always assume that
and all interfaces between different materials are sufficiently smooth
(e.g. piecewise from ).
We use the standard Sobolev spaces (cf. e.g., [7])
and equipped with the norm
and the scalar product ,
and the space
Next, we define the space with
where
and Einstein's summation convention is applied to indices and ,
and
It is a standard result that is a norm of
.
The following lemma (cf. [6]) allows us to represent the norm
of the
.
Lemma 1
For all
the equation

(16) 
holds.
Proof.
Assume
. We rewrite the
operator
as follows
and after having applied partial integration twice we arrive at

(17) 
Since
is dense in
,
equation (16) holds for all
.
Let us discuss further spaces, cf. [4].
We define
It is proved in [7] that
We remark that
has a zero trace of the normal component on
, whereas
has a zero trace of the tangential component.
Therefore, the space
seems to be appropriate to
the flux line boundary condition, cf. (12).
Since we are interested in formulating a gauge condition
that involves the divergence of the vector potential,
we introduce the space

(18) 
equipped with the norm

(19) 
>From [7], we have the following results.
Lemma 2
Assume that is bounded, Lipschitzcontinuous, simplyconnected
and has just one component.
Then there exists a positive constant such that
holds.
Proof. See [7], Lemma 3.4.
Lemma 3
Assume that is bounded, Lipschitzcontinuous, simplyconnected
and has just one component.
Then, the seminorm

(20) 
is a norm in Y, too.
Proof. Lemma 3 is a direct consequence of Lemma 2.
The next lemma requires more than the standard assumptions
for technical magnetic field problems.
Indeed, electromagnetic devices may have reentrant corners.
Therefore, we will not base our further investigations on Lemma 4.
Lemma 4
Assume further that either has a
boundary,
or is a convex polyhedron. Then is continuously
imbedded in .
Proof. See [7], Theorem 3.7.
