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

## Equivalence with magnetostatics

Next, we have to prove that the solution of the variational problem (23), (24) represents a solution of the magnetostatic problem, too. Indeed, in (23) we have added the terms and to the formulation of (15). We show that for the solution the sum of both terms vanishes. We recall the following theorem from [7].

THEOREM 3   Assume that is bounded, Lipschitz-continuous and simply-connected. A function satisfies

iff there exists a unique function such that

Proof. See [7], Theorem 2.9.
In the next theorem, we will assume that the electrical current density is divergence-free. Indeed, since electrical charges cannot appear or disappear, this assumption represents the physical behaviour.

THEOREM 4   Assume that is bounded, Lipschitz-continuous and simply-connected. Assume further that is divergence-free, i.e.,
 (34)

Then, the unique solution of

 (35) (36)

fulfills
 (37)

Proof. Let us choose an arbitrary . Then it is well-known that the Dirichlet problem

 (38) (39)

has a unique weak solution . >From Theorem 3 we deduce that there exists a with
 (40)

and
 (41)

Next, from (39) we get that the tangential derivatives of vanish, i.e. that the tangential components of are zero. Thus, holds. Further, we can determine the divergence of as follows
 (42)

Consequently, and . We can apply as a test function in (35), and we get with (40)
 (43) (44) (45)

With the assumptions (39) and (34) on and , the right-hand side is zero. Therefore,
 (46)

holds, and setting for any , we arrive at (37).
Thus, with (15) we get
 (47)

and we conclude the relation
 (48)

in the weak sense. We remark that we do not require that on . Further, setting , we get from (46) and (36) that

and . Therefore, we can omit the -term in (35), and solve the coercive problem:
(VF1) Find such that
 (49)

The unique solution of (VF1) coincides with the in the solution of (VF1). We may discretize and solve the problem in the formulation (VF1) instead of (VF1).
 HEJ, HU ISSN 1418-7108Manuscript no.: ANM-981030-A