HEJ, HU ISSN 1418-7108Manuscript no.: ANM-981030-A

## 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, Lipschitz-continuous, simply-connected 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, Lipschitz-continuous, simply-connected 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 re-entrant 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.
 HEJ, HU ISSN 1418-7108Manuscript no.: ANM-981030-A