The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B

31 Function SpacesIn this section we introduce function spaces necessary for the boundary integral formulationof boundary value problems for the Helmholtz equation. For a more detailed treatment ofthis topic we refer to [1] and [11].For d ∈ N we define a multiindexα = [α 1 , . . . , α d ] ∈ N d 0of the length |α| := α 1 + · · · + α d .u: R d → R can thus be expressed asPartial derivatives of a smooth enough functionD α u :=∂ |α|∂x α 11 . . . u.∂xα ddFor a complex-valued function u: R d → C we define the partial derivatives asD α u := D α (Re u) + iD α (Im u).1.1 Continuous FunctionsIn the following text we assume that Ω denotes a domain, i.e., a non-empty open connectedsubset of R d . By C(Ω) = C 0 (Ω) we denote the space of functions defined and continuousin Ω. For k ∈ N we denote by C k (Ω) the space of k times differentiable functions suchthat∀α, |α| ≤ k : D α u ∈ C(Ω).We also defineC ∞ (Ω) := k∈NC k (Ω).Moreover, for k ∈ N 0 ∪ {∞} we define the space C0 k (Ω) asu ∈ C0 k (Ω) ⇔ u ∈ C k (Ω) ∧ supp u := {x ∈ Ω : u(x) ≠ 0} ⊂ Ω is compact .By C(Ω) = C 0 (Ω) we denote the space of functions in C(Ω) that are bounded anduniformly continuous in Ω. Note that such functions are continuously extendable to ∂Ωasu(x) :=lim u(˜x)Ω∋˜x→x∈∂Ωfor x ∈ ∂Ωand in the following text we assume that these functions are already extended in sucha way. Similarly as above, C k (Ω) with k ∈ N represents the space of all functions inu ∈ C k (Ω) such that D α u ∈ C(Ω) for all α, |α| ≤ k. Equipped with the norm∥u∥ C k (Ω) := sup |D α u(x)|x∈Ω|α|≤k