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

6 1 Function SpacesAll L p (Ω) spaces with p ∈ [1, ∞) ∪ {∞} are Banach spaces. Moreover, L 2 (Ω) is aHilbert space with the inner product⟨u, v⟩ L 2 (Ω) := u(x)v(x) dxinducing the norm ∥ · ∥ L 2 (Ω). In particular, for v = u, i.e., for the square of the L 2 (Ω)norm of u we get the equality∥u∥ 2 L 2 (Ω) = ⟨u, u⟩ L 2 (Ω) = u(x)u(x) dx = |u(x)| 2 dx.ΩFurthermore, we introduce L 1 loc(Ω) as the space of locally integrable measurable functionsu: Ω → C, i.e., for such functions it holds|u(x)| dx < ∞ for all compact subsets K ⊂ Ω.KNote that every function f ∈ L 1 loc(Ω) can be identified with a distribution defined as⟨f, ϕ⟩ := f(x)ϕ(x) dx for all ϕ ∈ C0 ∞ (Ω).ΩA partial derivative of a distribution F is a distribution D α F defined byΩ⟨D α F, ϕ⟩ := (−1) |α| ⟨F, D α ϕ⟩ for all ϕ ∈ C ∞ 0 (Ω). (1.1)ΩSince we deal with the Helmholtz equation in the following sections, it is necessary tointroduce Sobolev spaces of the first order. We define W 1,p (Ω) asW 1,p (Ω) :=u ∈ L p (Ω):∂u∈ L p (Ω) for k ∈ {1, . . . , d} ,∂x kwhere the derivatives must be considered in the distributional sense. Hence, W 1,p (Ω) is asubspace of L p (Ω). We denote by W 1,p0 (Ω) the closure of C0 ∞(Ω) in the space W 1,p (Ω).Both previously introduced spaces are Banach spaces for p ∈ [1, ∞) ∪ {∞} with respect tothe norm d1/p∥u∥ W 1,p (Ω) := |u(x)| p +∂up (x)Ω∂x k dx. (1.2)According to Theorem 3.22 in [1], for Lipschitz domains it holds that the set of functionsin C ∞ 0 (Rd ) restricted to Ω is dense in W 1,p (Ω) and thus for every function u ∈ W 1,p (Ω)there exists a sequence (ϕ n ) ⊂ C ∞ 0 (Rd ) such thatk=1lim ∥ϕ n | Ω − u∥ W 1,p (Ω) = 0.