8]) to three dimensions.
Let be decomposed into subdomains
representing materials with different magnetic properties. We assume that the function in (8) depends on the position , but is always the same function in one of the , i.e.,
We define the bivariate form which is nonlinear in its first argument, but linear in its second, by
8), (9) can be written as:
(VF2) Find a pair such that
where the bilinear form is the identical as in (22). Now, we prove monotonicity and Lipschitz continuity of the form resp. the (nonlinear) operator defined by
Lemma 5 Assume (M1) and (M2). Then is strongly monotone, i.e., the inequalityProof. The proof is similar to the proof of Lemma 2.1 in . We consider vectors , and assume that holds without loss of generality. Then, we get
and from (M1) and (M2) it follows that
holds. Setting and and integrating we obtain
3 we get the estimate (57).
Lemma 6 Suppose that (M4) and (M5) hold. Then is Lipschitz-continuous, i.e., the inequalityProof. >From (M4) and (M5) we obtain
and by the mean value theorem we get
and with well-known estimates, an equation for the term, and Lemma 3 we obtain
and the desired estimate (59).