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

## The Brezzi theorem

We recall the Brezzi theorem, e.g. from [3].

THEOREM 1   Suppose a mixed variational problem in abstract setting
Find such that

 (25) (26)

Assume that the following conditions are satisfied
(A1) , ,
(A2) and with

(A3) the LBB (Ladyzenskaya-Babuška-Brezzi) condition, i.e.
such that
 (27)

(A4) the - ellipticity, i.e. with
 (28)

where
 (29)

Then there exists a unique solution and the a-priori estimates

hold.

Proof. See, e.g., [4].
We apply Theorem 1 to the variational formulation (VF1).

THEOREM 2   Suppose that , and that is a piecewise constant function fulfilling
 (30)

Then (23), (24) has a unique solution for which a-priori estimates

hold.

Proof. We have to verify the assumptions of Theorem 1.
(A1) It is the assumption, and .
(A2) Applying Lemma 3 we get

i.e., we can set , and

i.e. .
(A3) The LBB condition has been proved for the Stokes problem, cf. e.g. [6, Theorem 3.7]. There exists a constant with
 (31)

>From Lemma 1 and (20) we get , and with it follows that
 (32)

(A4) We can prove ellipticity for the whole space , i.e.

 (33)

and .
We remark that it may be useful to choose depending on the actual coefficients , e.g., .
 HEJ, HU ISSN 1418-7108Manuscript no.: ANM-981030-A