Integral Equations

14.2 Generalized potentialsLet S Π (Γ) ∈ R 2 be a domain bounded by the closed piecewise smooth contour Γ. We assumethat a rectilinear interval Γ 0 is a subset of Γ, so that Γ 0 = {x : y = 0, x ∈ [a, b]}.Let us say that functions U l (x) are the generalized single layer (SLP) (l = 1) or double layer(DLP) (l = 2) potentials ifwhere∫U l (x) = K l (x, t)ϕ l (t)dt,Γx = (x, y) ∈ S Π (Γ),K l (x, t) = g l (x, t) + F l (x, t) (l = 1, 2),g 1 (x, t) = g(x, y 0 ) = 1 π ln 1|x − y 0 | , g 2(x, t) = ∂∂y 0g(x, y 0 ) [y 0 = (t, 0)],F 1,2 are smooth functions, and we shall assume that for every closed domain S 0Π (Γ) ⊂ S Π (Γ),the following conditions holdi) F 1 (x, t) is once continuously differentiable with respect to thevariables of x and continuous in t;ii) F 2 (x, t) andare continuous.F 1 2 (x, t) = ∂ ∂y∫tqF 2 (x, s)ds, q ∈ R 1 ,We shall also assume that the densities of the generalized potentials ϕ 1 ∈ L (1)2 (Γ) andϕ 2 ∈ L (2)2 (Γ), where functional spaces L (1)2 and L (2)2 are defined in Section 3.15 Reduction of boundary value problems to integralequationsGreen’s formulas show that each harmonic function can be represented as a combination ofsingle- and double-layer potentials. For boundary value problems we will find a solution in theform of one of these potentials.Introduce integral operators K 0 and K 1 acting in the space C(Γ) of continuous functionsdefined on contour Γand∫K 0 (x) = 2Γ∫K 1 (x) = 2Γ∂Φ(x, y)∂n yϕ(y)dl y , x ∈ Γ (295)∂Φ(x, y)∂n xψ(y)dl y , x ∈ Γ. (296)The kernels of operators K 0 and K 1 are continuous on Γ As seen by interchanging the orderof integration, K 0 and K 1 are adjoint as integral operators in the space C(Γ) of continuousfunctions defined on curve Γ.56