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84 2. Boundary Integral Equations

⎞ ⎟ ⎟ ⎟ ⎟ ⎠ 1 σ 2 (x) (2.4.22) (2.4.23)

(0,σ 2 )(z) (0,σ 2 )(z) (0,σ 2 )(z) − 2

V ·∇ z V M z V (0,σ 2 )(z)
n x
N z V ⎞ ⎟ ⎟ ⎠

1 σ 1 (x)

(σ 1 , 0)(z) (σ 1 , 0)(z) − 2 (σ 1 , 0)(z) | Γ . V 14 V 24 V 34 1 I K 44 + 2

V ·∇ z V (σ 1 , 0)(z) N z V K 33
n x ⎟ ⎟ ⎠ by V 13 V 23 D 43
M z V V 14 ⎞ V 24 ⎟ V 34 ⎟ K 44 ∂u ,Mu,Nu 1 I − 2

1 ϕ 2 (x) V 13 V 23 −K 33 D 43 u, ∂n defined is V 12 K 22 D 32 D 42

− 2 V 12 K 22 D 32 D 42 = 1 I + 2
−W(0,ϕ 2 )(z) ·∇ z W(0,ϕ 2 )(z) −M z W(0,ϕ 2 )(z) −N z W(0,ϕ 2 )(z) ⎛ −K 11 ⎜ D 21 ⎜ ⎜ ⎜ D 31 ⎝ D 41 =(ϕ 1 ,ϕ 2 ,σ 1 ,σ 2 ) bi–Laplacian the ⎛ 1 I K 11 − ⎜ 2 D 21 D 31 D 41

−n x = ϕ σ with ⎜ = K + ⎝

1 ϕ 1 (z) associated 1 I := 2

− 2 ·∇ z W(ϕ 1 , 0)(z) −M z W(ϕ 1 , 0)(z) −N z W(ϕ 1 , 0)(z) projector C Ω
⎛ −W(ϕ 1 , 0)(z) −n x Calderon

= ⎜ ⎜ ⎜ ⎜ ⎝ write we the
ϕ σ lim Ω z→x∈Γ where

K Then