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

9Note that the trace operator is not surjective, i.e., there exist functions in L 2 (∂Ω) thatare not traces of any function in Hloc 1 (Ω). Therefore, we introduce H1/2 (∂Ω) as the tracespace of Hloc 1 (Ω), i.e.,H 1/2 (∂Ω) := γ 0 (Hloc 1 (Ω)).Obviously, H 1/2 (∂Ω) is a linear subset of L 2 (∂Ω). Equipped with the Sobolev–Slobodeckiinorm ∥u∥ H 1/2 (∂Ω) := ∥u∥ 2 L 2 (∂Ω) + |u(x) − u(y)| 2 1/2∥x − y∥ 3 ds x ds y∂Ωthe space is complete. Note that for a bounded domain Ω we have an equivalent norm∥u∥ H 1/2 (∂Ω) :=We define H −1/2 (∂Ω) as the dual space to H 1/2 (∂Ω), i.e., ∗H −1/2 (∂Ω) := H 1/2 (∂Ω)with the standard supremum norm∂Ωinf ∥v∥ H 1 (Ω). (1.5)v∈H 1 (Ω)γ 0 v=u∥f∥ H −1/2 (∂Ω) :=supu∈H 1/2 (∂Ω)u≠0|⟨f, u⟩|∥u∥ H 1/2 (∂Ω). (1.6)For a relatively open part of the boundary Γ ⊂ ∂Ω we define the spacesH 1/2 (Γ ) := v = ṽ| Γ : ṽ ∈ H 1/2 (∂Ω) ,H 1/2 (Γ ) := v = ṽ| Γ : ṽ ∈ H 1/2 (∂Ω), supp ṽ ⊂ Γ , ∗H −1/2 (Γ ) := H 1/2 (Γ ),∗H −1/2 (Γ ) := H 1/2 (Γ ).These spaces will be used for the purposes of mixed boundary value problems. For a moredetailed treatment on this topic see [12] or [18].1.4 Generalized Normal DerivativesLet us now assume that Ω denotes a bounded domain. Recall that for a function u ∈ H 1 (Ω)there exists a unique trace γ 0 u ∈ H 1/2 (∂Ω) generalizing the notion of a restriction tothe boundary. To generalize a normal derivative in the same way we would need higherwould have to be inH 1 (Ω). In the following section we will, however, show that it is possible to introducenormal derivatives for functions in H 1 (Ω, ∆ + κ 2 ).regularity of u. Namely, the distributional partial derivatives ∂u∂x k