Integral Equations

which proves the continuity of the kernel (270).Statement 2 (Gauss formula) Let D ∈ R 2 be a domain bounded by the closed smoothcontour Γ. For the double-layer potential with a constant density∫v 0 (x) =Γ∂Φ(x, y)∂n ydl y , Φ(x, y) = 12π ln 1|x − y| , (272)where the (exterior) unit normal vector n of Γ is directed into the exterior domain R 2 \ ¯D, wehavev 0 (x) = −1, x ∈ D,Proof follows for x ∈ R 2 \ ¯D from the equalityv 0 (x) = − 1 , x ∈ Γ, (273)2v 0 (x) = 0, x ∈ R 2 \ ¯D.∫Γ∂v(y)∂n ydl y = 0 (274)applied to v(y) = Φ(x, y). For x ∈ D it follows from Green’s third formula∫ (u(x) =ΓΦ(x, y) ∂u∂n y− u(y))∂Φ(x, y)dl y , x ∈ D, (275)∂n yapplied to u(y) = 1 in D.Note also that if we setv 0 (x ′ ) = − 1 2 , x′ ∈ Γ,we can also write (273) asv 0 ±(x ′ ) = limh→±0 v(x + hn x ′) = v0 (x ′ ) ± 1 2 , x′ ∈ Γ. (276)Corollary. Let D ∈ R 2 be a domain bounded by the closed smooth contour Γ. Introducethe single-layer potential with a constant density∫u 0 (x) =For the normal derivative of this single-layer potentialΓΦ(x, y)dl y , Φ(x, y) = 12π ln 1|x − y| . (277)∂u 0 ∫(x)=∂n xΓ∂Φ(x, y)∂n xdl y , (278)53