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

## Nonlinear extension of Brezzi's theorem

The following lemma provides equivalent formulations for the LBB condition in an abstract setting.

Lemma 7   Define the linear operator and its adjoint operator in abstract spaces , by
 (62) (63)

Then, the following formulations of the LBB condition , , are equivalent.
such that
 (64)

The operator is an isomorphism, and the estimate
 (65)

holds, where
The operator is an isomorphism, and the estimate
 (66)

holds, where denotes the polar of , i.e.

Proof. The lemma is proved, e.g., in [3, Lemma 4.2].
Next, we present an extension of Brezzi's theorem to a class of nonlinear problems.

THEOREM 5   Suppose a mixed variational problem in abstract setting
Find such that

 (67) (68)

where the bivariate form is assumed to be linear with respect to its second argument only, and is a bilinear form. Assume that the following conditions are satisfied
(A1) ,
(A2) with

(A3) the LBB (Ladyzenskaya-Babuška-Brezzi) condition, i.e.
such that
 (69)

(A4) the strong - monotonicity of , i.e. with
 (70)

where
 (71)

(A5) and the Lipschitz continuity of , i.e., with

Then there exists a unique solution .

Proof. >From (A4) and (A5) it follows that the nonlinear problem
Find with
 (72)

has a unique solution (see, e.g., [19, Theorem 24.2.]). Then we define by
 (73)

Since

we get , and with Lemma 7 () it follows that there exists a unique with
 (74)

Finally, we conclude from , (73), (74), that the pair fulfills (67), (68).
Now, we apply Theorem 5 to the variational formulation (VF2).

THEOREM 6   Suppose that and that the material function fulfills (50) and (M1), (M2), (M4) and (M5). Then (55), (56) have a unique solution.

Proof. The assumptions (A1), (A2) and (A3) are verified in the proof of Theorem 2. Further, (A4) and (A5) follow from the Lemmata 5 and 6.
The results of Section 3.4 can be carried over to the nonlinear case without modification.

THEOREM 7   Assume that is bounded, Lipschitz-continuous and simply-connected. Assume further that is divergence-free, i.e.,
 (75)

Then, the unique solution of

fulfills
 (76)

Proof. Coincides with that of Theorem 4. Indeed, the only term in (35) involving is not changed during the proof of Theorem 4.
Thus, the variational formulation of the nonlinear problem is adequate to the physical problem. Again, we have such that we can omit the -term and solve the coercive problem:
(VF2) Find such that
 (77)

The unique solution of (VF2) coincides with the in the solution of (VF2).
 HEJ, HU ISSN 1418-7108Manuscript no.: ANM-981030-A