ABSTRACT
The goal of this chapter is simple: we want to build a theory that allows us to prove the well-posedness of the Neumann problem for the Laplace operator − Δ u = f , ∂ n u = h . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/umath7_1.jpg"/> The solution to this problem is clearly nonunique (all the operators in the left-hand side vanish when applied to constant functions) and it is clear that for the problem to have a unique solution we need 〈 h , 1 〉 Γ = 〈 ∂ n u , γ 1 〉 Γ = ( Δ u , 1 ) Ω = − ( f , 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/umath7_2.jpg"/> that is, the data have to be compatible. Proving some sort of coercivity condition will require us to show the Poincaré inequality inf c ∈ R ∥ u − c ∥ Ω ≤ C ∥ ∇ u ∥ Ω ∀ 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/umath7_3.jpg"/> We will be able to transform this inequality into other similar (and equivalent) ones that prove coercivity of the Dirichlet form ( ∇ u , ∇ v ) 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-math7_1.jpg"/> in different subspaces of H 1(Ω). However, the proof of the Poincaré inequality that we will give requires the concept of compactness and, specifically, the compact embedding of H 1(Ω) into L 2(Ω).
