Integral Equations

where the (exterior) unit normal vector n x of Γ is directed into the exterior domain R 2 \ ¯D, wehave∂u 0 (x)∂n x= 1, x ∈ D,∂u 0 (x)∂n x= 1 ,2x ∈ Γ, (279)∂u 0 (x)∂n x= 0, x ∈ R 2 \ ¯D.Theorem 35 Let D ∈ R 2 be a domain bounded by the closed smooth contour Γ. The doublelayerpotential∫∂Φ(x, y)v(x) =ϕ(y)dl y , Φ(x, y) = 1∂n y 2π ln 1|x − y| , (280)Γwith a continuous density ϕ can be continuously extended from D to ¯D and from R 2 \ ¯D toR 2 \ D with the limiting values on Γor∫v ± (x ′ ) =Γ∂Φ(x ′ , y)∂n yϕ(y)dl y ± 1 2 ϕ(x′ ), x ′ ∈ Γ, (281)v ± (x ′ ) = v(x ′ ) ± 1 2 ϕ(x′ ), x ′ ∈ Γ, (282)wherev ± (x ′ ) = lim v(x + hn x ′). (283)h→±0Proof.1. Introduce the function∫I(x) = v(x) − v 0 (x) =Γ∂Φ(x, y)∂n y(ϕ(y) − ϕ 0 )dl y (284)where ϕ 0 = ϕ(x ′ ) and prove its continuity at the point x ′ ∈ Γ. For every ɛ > 0 and every η > 0there exists a vicinity C 1 ⊂ Γ of the point x ′ ∈ Γ such that|ϕ(x) − ϕ(x ′ )| < η, x ′ ∈ C 1 . (285)SetThen∫I = I 1 + I 2 = . . . +C 1|I 1 | ≤ ηB 1 ,∫Γ\C 1. . . .54