Sobolev Spaces

In this chapter we use the notations D = (D ₁,…, D_n) = (∂/∂x ₁,…, ∂/∂x_n) and D α = D 1 α 1 ⋯ D n α n , | α | = α 1 + ⋯ + α n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755419/25763ea9-efbc-4fdd-b209-84209bf4ed4e/content/inequ3_1.tif"/> for a vector α = (α ₁,…, α_n) with integral components α_i ≥ 0. We often write D^m to denote mth order derivatives, i.e. D^m is one of D^α with |α| = m. Let Ω be a nonempty open subset of Rⁿ and m be a nonnegative integer. Then C^m (Ω) denotes the set of all functions whose derivatives of order up to m are all continuous in Ω, and C 0 m ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755419/25763ea9-efbc-4fdd-b209-84209bf4ed4e/content/inequ3_2.tif"/> the totality of functions belonging to C^m (Ω) and with compact support in Ω. We denote by B^m (Ω) the set of all functions which are bounded and continuous in fi together with their derivatives of order up to m. For 0 < h < 1 we denote by B^m+h (Ω) the set of all functions belonging to B^m (Ω) whose mth order derivatives are all uniformly Hölder continuous in Ω with exponent h. Similarly the sets B m ( Ω ¯ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755419/25763ea9-efbc-4fdd-b209-84209bf4ed4e/content/inequ3_3.tif"/> and B m + h ( Ω ¯ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755419/25763ea9-efbc-4fdd-b209-84209bf4ed4e/content/inequ3_4.tif"/> are defined replacing Ω by Ω ¯   https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755419/25763ea9-efbc-4fdd-b209-84209bf4ed4e/content/inequ3_5.tif"/> . For u ∈ B^m (Ω) we put