Properties of Sobolev Spaces
This chapter collects a number of important properties of Sobolev spaces. Almost every claim is provided together with a proof. The main result is the density of \(\mathcal D(\overline \varOmega )\) in Ws, p(Ω) and proved in Theorem 3.4.5. The necessary tools to establish these proofs are introduced and intermediate results are presented in the following subsections. This entire chapter can be viewed as a preparation for subsequent ones on traces, Chap. 4, and to meaningfully define the weak forms of some partial differential equations in Chap. 5.
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