ABSTRACT
In this chapter we give a weak interpretation of the normal derivative ∇ u ⋅ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math6_1.jpg"/> for u ∈ H 1 ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math6_2.jpg"/> with Δ u ∈ L 2 ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math6_3.jpg"/> . This will be done using duality on the trace space H 1 / 2 ( Γ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math6_4.jpg"/> (as usual Γ := ∂ Ω https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math6_5.jpg"/> ) and Green’s first identity as the definition of the normal derivative. The process will be done gradually by first working on what we understand by the normal component on the boundary of a vector field p ∈ L 2 ( Ω ) := L 2 ( Ω ; R 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-math6_6.jpg"/> such that ∇ ⋅ p ∈ L 2 ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math6_7.jpg"/> . We will use the definition of the weak normal derivative to explore Neumann boundary conditions on several coercive problems. The Neumann problem for the Laplacian − Δ u = f in Ω , ∇ u ⋅ n = h on Γ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/umath6_1.jpg"/> will have to wait until Chapter 7 and so will some problems for which it is less clear that the associated bilinear form is coercive in the entire space H 1 ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math6_8.jpg"/> . The reason for this postponement is the need to prove a family of Poincaré type inequalities, which are derived from some compact embeddings.
