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

13In the previous paragraphs we showed how to generalize the concept of a normal derivativeof a function in H 1 (Ω, ∆ + κ 2 ) with a bounded domain Ω. However, it is also possibleto introduce the Neumann trace operator γ 1 for functions defined on an unbounded domain.Since the trace is only dependable on the behaviour of the function in the vicinityof the boundary, we haveγ 1 : H 1 loc (Ω, ∆ + κ2 ) → H −1/2 (∂Ω).Because H 1 loc (Ω, ∆+κ2 ) coincides with H 1 (Ω, ∆+κ 2 ) for bounded domains, the precedingdefinitions agree with this concept. For a more detailed treatment of this topis see [15],Section 2.7.Note that for a function u ∈ H 2 (Ω), whereH 2 (Ω) := u ∈ L 2 (Ω): D α u ∈ L 2 (Ω) for |α| ≤ 2 ,we have⟨γ 1 u, γ 0 v⟩ =dk=1∂Ωγ 0 ∂u(x)n k (x)γ 0 ∂uv(x) ds =∂x k ∂Ω ∂n (x)γ0 v(x) ds.