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

## Monotonicity and Lipschitz continuity

In 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 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.,
 (50)

We suppose the following properties of the functions which are justified by the physical model (cf. [8]):
(M0)
(M1)
(M2)
is a monotonic increasing function,
(M3)
where for ferromagnetic materials,
(M4)
there exists and there exists a constant with
(M5)
there exists a constant with
Let and be the global constants with

 (51) (52) (53)

We define the bivariate form which is nonlinear in its first argument, but linear in its second, by
 (54)

Then, the variational formulation for the problem (8), (9) can be written as:
(VF2) Find a pair such that

 (55) (56)

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 inequality
 (57)

holds.

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
 (58)

and Lemma 3 we get the estimate (57).

Lemma 6   Suppose that (M4) and (M5) hold. Then is Lipschitz-continuous, i.e., the inequality
 (59)

holds.

Proof. >From (M4) and (M5) we obtain

and by the mean value theorem we get
 (60)

Consider vectors . The same calculation as in the proof of Lemma 2.2 in [8] yields
 (61)

We set multiply with , and take the integral over . The result is

and with well-known estimates, an equation for the term, and Lemma 3 we obtain

and the desired estimate (59).
 HEJ, HU ISSN 1418-7108Manuscript no.: ANM-981030-A