represents a solution of the magnetostatic problem, too.
Indeed, in (23) we have added the terms
to the formulation of (15).
We show that for the solution
the sum of both terms vanishes.
We recall the following theorem from .
THEOREM 3 Assume that is bounded, Lipschitz-continuous and simply-connected. A function satisfiesProof. See , Theorem 2.9.
iff there exists a unique function such that
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.,Proof. Let us choose an arbitrary . Then it is well-known that the Dirichlet problem
has a unique weak solution . >From Theorem 3 we deduce that there exists a with
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 right-hand side is zero. Therefore,
Thus, with (15) we get
and . Therefore, we can omit the -term in (35), and solve the coercive problem:
(VF1) Find such that