The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B
The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B
4 1 Function Spacesthe space C k (Ω) is a Banach space. Furthermore, we define C ∞ (Ω) as the space offunctions in C ∞ (Ω) that are continuously extendable to ∂Ω. Note that functions in thisspace do not have to be bounded nor uniformly continuous.For u ∈ C k (Ω) with k ∈ N 0 , a multiindex α, |α| ≤ k and λ ∈ (0, 1] we denote|D α u(x) − D α u(y)|H α,λ (u) := supx,y∈Ω ∥x − y∥ λ .x≠yWe define the space of Hölder continuous functions asC k,λ (Ω) := u ∈ C k (Ω): H α,λ (u) < ∞ for all α, |α| = k .Together with the norm∥u∥ C k,λ (Ω) := x∈Ω|α|≤ksup |D α u(x)| + |α|=k|D α u(x) − D α u(y)|supx,y∈Ω ∥x − y∥ λx≠ythe space C k,λ (Ω) is a Banach space. Setting k = 0 and λ = 1 we get the space C 0,1 (Ω)of Lipschitz continuous functions in Ω.Definition 1.1. A domain Ω ⊂ R d with a compact boundary ∂Ω is a C k,λ domain if thereexists a finite family of open sets {U i } n i=1 such that for every i ∈ {1, . . . , n} there exist• a Cartesian system of coordinates• ε i , δ i ∈ R + ,• a function a i : R d−1 → Rsatisfying(y i 1, . . . , y i d−1 , yi d ) = (yi , y i d ), where yi := (y i 1, . . . , y i d−1 ),• Γ i := U i ∩ ∂Ω = {(y i , y i d ): ∥yi ∥ < δ i , y i d = a i(y i )},• U + i:= {(y i , y i d ): ∥yi ∥ < δ i , a i (y i ) < y i d < a i(y i ) + ε i } ⊂ Ω,• U − i:= {(y i , y i d ): ∥yi ∥ < δ i , a i (y i ) − ε i < y i d < a i(y i )} ⊂ R d \ Ω,• a i ∈ C k,λ ({y i : ∥y i ∥ ≤ δ i }).For a Lipschitz domain, i.e., a C 0,1 domain (for an example of a Lipschitz domain inR 2 see Figure 1.1), the unit outward normal vector n = (n 1 , . . . , n d ) is defined almosteverywhere on ∂Ω. The coordinates n 1 , . . . , n d are bounded measurable functions on ∂Ω.Note that in this case the term ‘measurable’ corresponds to a surface measure defined on∂Ω. This abuse of terminology could be treated by introducing mappings from U i ∩ ∂Ω tothe global coordinate system and the measure could be understood as a (d−1) dimensionalLebesgue measure. However, throughout this text we use the simpler notation and referto this interpretation.
5∂Ωy i 2ΩU + iU − iΓ iy i 1Figure 1.1: Lipschitz domain in R 2 .1.2 Lebesgue and Sobolev SpacesFor a domain Ω ⊂ R d and p ∈ [1, ∞) we introduce L p (Ω) as the space of measurablefunctions u: Ω → C with1/p∥u∥ L p (Ω) := |u(x)| dx p < ∞.ΩRemark 1.2. In L p (Ω) we identify functions that are equal almost everywhere in Ω, thusthe elements of L p (Ω) are actually equivalence classes. By the relation u ∈ L p (Ω) weunderstand that there exists an equivalence class in L p (Ω) such that u belongs to it.For functions u ∈ L p (Ω) and v ∈ L q (Ω) with1p + 1 q= 1, p, q ∈ (1, ∞)it holds that uv ∈ L 1 (Ω) and the Hölder inequality|u(x)v(x)| dx ≤ ∥u∥ L p (Ω)∥v∥ L q (Ω)is satisfied.ΩThe L ∞ (Ω) space is defined as the space of measurable functions u: Ω → C satisfying∥u∥ L ∞ (Ω) := ess sup |u| :=Ωwhere µ d denotes the Lebesgue measure in R d .infE⊂Ωµ d (E)=0sup |u(x)| < ∞,x∈Ω\E
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5∂Ωy i 2ΩU + iU − iΓ iy i 1Figure 1.1: Lipschitz domain in R 2 .1.2 Lebesgue and Sobolev SpacesFor a domain Ω ⊂ R d and p ∈ [1, ∞) we introduce L p (Ω) as <strong>the</strong> space of measurablefunctions u: Ω → C with1/p∥u∥ L p (Ω) := |u(x)| dx p < ∞.ΩRemark 1.2. In L p (Ω) we identify functions that are equal almost everywhere in Ω, thus<strong>the</strong> elements of L p (Ω) are actually equivalence classes. By <strong>the</strong> relation u ∈ L p (Ω) weunderstand that <strong>the</strong>re exists an equivalence class in L p (Ω) such that u belongs to it.For functions u ∈ L p (Ω) and v ∈ L q (Ω) with1p + 1 q= 1, p, q ∈ (1, ∞)it holds that uv ∈ L 1 (Ω) and <strong>the</strong> Hölder inequality|u(x)v(x)| dx ≤ ∥u∥ L p (Ω)∥v∥ L q (Ω)is satisfied.Ω<strong>The</strong> L ∞ (Ω) space is defined as <strong>the</strong> space of measurable functions u: Ω → C satisfying∥u∥ L ∞ (Ω) := ess sup |u| :=Ωwhere µ d denotes <strong>the</strong> Lebesgue measure in R d .infE⊂Ωµ d (E)=0sup |u(x)| < ∞,x∈Ω\E
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