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, Lipschitzcontinuous and simplyconnected.
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 divergencefree.
Indeed, since electrical charges cannot appear or disappear,
this assumption represents the physical behaviour.
THEOREM 4
Assume that is bounded, Lipschitzcontinuous and simplyconnected.
Assume further that
is divergencefree,
i.e.,

(34) 
Then, the unique solution
of
fulfills

(37) 
Proof.
Let us choose an arbitrary
.
Then it is wellknown that the Dirichlet problem
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
Consequently,
and
.
We can apply as a test function in
(35), and we get with (40)
With the assumptions
(39) and (34) on and ,
the righthand 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).
