 
Monotonicity and Lipschitz continuityIn many practical cases, the nonlinear behaviour of ferromagnetic materials cannot be neglected. Therefore, we extend the analysis of 2D monotone, Lipschitz continuous nonlinear problems (see [8]) to three dimensions. Let be decomposed into subdomainsrepresenting 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 suppose the following properties of the functions which are justified by the physical model (cf. [8]):
We define the bivariate form which is nonlinear in its first argument, but linear in its second, by Then, the variational formulation for the problem (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 Proof. The proof is similar to the proof of Lemma 2.1 in [8]. 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 and with and Lemma 3 we get the estimate (57).
Lemma 6
Suppose that (M4) and (M5) hold.
Then is Lipschitzcontinuous, i.e., the
inequality
Proof.
>From (M4) and (M5) we obtain
holds. and by the mean value theorem we get Consider vectors . The same calculation as in the proof of Lemma 2.2 in [8] yields We set multiply with , and take the integral over . The result is and with wellknown estimates, an equation for the term, and Lemma 3 we obtain and the desired estimate (59).  
