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

313.1 Single Layer Potential OperatorLet us first consider the operator V κ defined by (3.3). Recall, that for the Dirichlet traceoperator γ 0 we haveγ 0 : H 1 loc (Ω) → H1/2 (∂Ω).Combining this operator with V κ we obtain the single layer potential operatorV κ : H −1/2 (∂Ω) → H 1/2 (∂Ω), V κ := γ 0 Vκ .From linearity and continuity of γ 0 and V κ we get that the single layer potential operatoris linear and continuous, i.e., there exists a constant c ∈ R + such that∥V κ s∥ H 1/2 (∂Ω) ≤ c∥s∥ H −1/2 (∂Ω)for all s ∈ H −1/2 (∂Ω).Theorem 3.8. The single layer potential operator V κ : H −1/2 (∂Ω) → H 1/2 (∂Ω) is coercive.Proof. From Theorem 6.22 in [18] we have that he operator V 0 corresponding to the Laplaceequation, i.e., the Helmholtz equation with κ = 0, is H −1/2 (∂Ω)-elliptic. Moreover, theoperator C := V 0 − V κ : H −1/2 (∂Ω) → H 1/2 (∂Ω) is compact (see [18], Section 6.9). Thus,we have⟨(V κ + C)s, s⟩ = ⟨V 0 s, s⟩ ≥ c∥s∥ 2 H −1/2 (∂Ω)for all s ∈ H −1/2 (∂Ω),which completes the proof.Theorem 3.9. For s ∈ L ∞ (∂Ω) there holds the representation(V κ s)(x) = v κ (x, y)s(y) ds y for x ∈ ∂Ω.Proof. The proof is similar to the proof of Lemma 6.7 in [18].∂ΩMoreover, for the jump of the Dirichlet trace of the single layer potential V κ s on theboundary we have[γ 0 Vκ s] := γ 0,ext Vκ s − γ 0,int Vκ s = 0 for all s ∈ H −1/2 (∂Ω). (3.5)3.2 Adjoint Double Layer Potential OperatorIn Section 1.4 we introduced the Neumann trace operatorγ 1 : H 1 loc (Ω, ∆ + κ2 ) → H −1/2 (∂Ω).