Generalised ridged domains
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The need for considering Sobolev spaces on scales of general function spaces to fine-tune classical results was demonstrated in Chapter 3. In Chapter 4 it was made clear that the nature of the boundary of the underlying domain Ω has considerable significance in many problems, and therefore a precise description of the boundary is essential for a detailed investigation. The main goal of this chapter is to study embeddings E = E(Ω): W(X,Y)(Ω) → Z(Ω), where X = X(Ω),Y = Y(Ω) and Z = Z(Ω) are Banach function spaces defined on a domain Ω ⊂ ℝn, W(X, Y) (Ω) denotes the Sobolev space defined in Section 3.6.1 (with k = 1), and the domain Ω is what we call a generalised ridged domain, GRD for short. We shall be mainly concerned with embedding theorems and the measure of non-compactness of E(Ω). The analysis in Section 4.2 indicated that the measure of non-compactness of the embedding map E(Ω) is determined by the nature of the boundary of Ω. For GRDs we can go a step further and highlight specific subsets of the boundary of Ω which are responsible for any non-compactness. Furthermore, two-sided estimates can be derived for the measure of non-compactness α(E) which determine whether or not α(E) is 0 (and hence E is compact) or is less than 1. To obtain such estimates which are also manageable in examples was a guiding principle behind the introduction of the GRDs in , . It is a very wide class of domains which includes horns, spirals, “rooms and passages” and ones with fractal boundaries like the Koch snowflake. A characteristic feature of a GRD is a central axis called a generalised ridge. This was introduced in  and was modelled on the ridge, central set or skeleton of an open subset of ℝn. This section will be concerned with the definitions and properties of these sets.
KeywordsMeasurable Subset Lorentz Space Finite Rank Banach Function Space Filter Base
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