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

47with an unbounded domain Ω ext := R 3 \ Ω and boundary conditions g D ∈ H 1/2 (Γ D ),g N ∈ H −1/2 (Γ N ). In the smooth case the solution to (3.53) is given byu(x) = − γ 1,ext u(y) v κ (x, y) ds y − g N (y) v κ (x, y) ds yΓ D Γ N+ g D (y) ∂v κ(x, y) ds y + γ 0,ext u(y) ∂v κ(x, y) ds y for x ∈ Ω extΓ D∂n y Γ N∂n ywith unknown Cauchy data γ 0,ext u| ΓN and γ 1,ext u| ΓD . Similarly as for the interior mixedproblem we have the boundary integral equations(V κ γ 1,ext u)(x) = − 1 2 γ0,ext u(x) + (K κ γ 0,ext u)(x) for x ∈ Γ D ⊂ ∂Ω,(D κ γ 0,ext u)(x) = − 1 2 γ1,ext u(x) − (K ∗ κγ 1,ext u)(x) for x ∈ Γ N ⊂ ∂Ω.(3.54)Inserting the functions t, s defined by (3.39) into (3.54) yields(V κ s)(x) − (K κ t)(x) = − 1 2 ˜g D(x) + (K κ˜g D )(x) − (V κ˜g N )(x) for x ∈ Γ D ,(K ∗ κs)(x) + (D κ t)(x) = − 1 2 ˜g N(x) − (K ∗ κ˜g N )(x) − (D κ˜g D )(x) for x ∈ Γ N(3.55)and the equivalent variational formulationa(s, t, q, r) = F (q, r) for all q ∈ H −1/2 (Γ D ), r ∈ H 1/2 (Γ N ) (3.56)with unknown functions s ∈ H −1/2 (Γ D ), t ∈ H 1/2 (Γ N ), the bilinear forma(s, t, q, r) := ⟨V κ s, q⟩ ΓD − ⟨K κ t, q⟩ ΓD + ⟨K ∗ κs, r⟩ ΓN + ⟨D κ t, r⟩ ΓNand the right-hand sideF (q, r) := − 1 2 I + K κ ˜g D , q − ⟨V κ˜g N , q⟩ ΓD + − 1 Γ D2 I − K∗ κ ˜g N , r − ⟨D κ˜g D , r⟩ ΓN .Γ NTheorem 3.31. If u ∈ H 1 loc (Ωext , ∆ + κ 2 ) is a solution to the mixed problem (3.53), thenthe functions s, t satisfy the system of boundary integral equations (3.55) and u has therepresentationu(x) = −( V κ (s + ˜g N ))(x) + (W κ (t + ˜g D ))(x) for x ∈ Ω ext . (3.57)Conversely, if s, t satisfy the system of boundary integral equations (3.55), then therepresentation formula (3.57) defines a solution u ∈ H 1 loc (Ωext , ∆ + κ 2 ) to the interiormixed problem (3.53).