ABSTRACT
This chapter deals with variational problems with the following mixed structure ( u , p ) ∈ V × M , a ( u , v ) + b ( v , p ) = ℓ ( v ) ∀ v ∈ V , b ( u , q ) = χ ( q ) ∀ q ∈ M , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/umath10_1.jpg"/> where V and M are Hilbert spaces, a and b are bounded bilinear forms, and ℓ ∈ V ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math10_1.jpg"/> , χ ∈ M ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math10_2.jpg"/> are generic data. We will start by studying the (not unique) solvability of the equation b ( u , q ) = χ ( q ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math10_3.jpg"/> and by discussing different forms of surjectivity of the associated operator B :V → M in terms of the bilinear form. This will just be a rewritten form of a corollary of the Banach closed range theorem. We will then state the necessary and sufficient Babuška-Brezzi conditions for the well-posedness of problems with mixed structure and relate some of them to constrained minimization problems. The rest of the chapter is devoted to classical examples of problems with mixed structure: Stokes, Darcy, and Brinkman flow, or a two-field formulation of the Reissner-Mindlin plate equations. The Stokes problem brings along the interesting and important nontrivial issue of discovering the range of the divergence operator when restricted to H 0 1 ( Ω ) := H 0 1 ( Ω ) d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math10_4.jpg"/> . This is actually related to the result which we left unproved in Section 7.6 on the closedness of the range of the gradient operator restricted to L 2(Ω). We will give more details about this problem 210and show how the result can be derived from a (still nontrivial) result on right inverses for the divergence operator. Finally, we will use the Brinkman and Reissner-Mindlin models to show how the solutions of some parameter-dependent model equations converge weakly to the solution of the reduced limit model, of which they are a singular perturbation.
