Integral Equations

14.1 Properties of potentialsStatement 1. Let D ∈ R 2 be a domain bounded by the closed smooth contour Γ. Then thekernel of the double-layer potentialV (x, y) =∂Φ(x, y)∂n y, Φ(x, y) = 12π ln 1|x − y| , (270)is a continuous function on Γ for x, y ∈ Γ.Proof. Performing differentiation we obtainV (x, y) = 1 cos θ x,y2π |x − y| , (271)where θ x,y is the angle between the normal vector n y at the integration point y = (x 0 , y 0 ) andvector x − y. Choose the origin O = (0, 0) of (Cartesian) coordinates at the point y on curveΓ so that the Ox axis goes along the tangent and the Oy axis, along the normal to Γ at thispoint. Then one can write the equation of curve Γ in a sufficiently small vicinity of the point yin the formy = y(x).The asumption concerning the smoothness of Γ means that y 0 (x 0 ) is a differentiable functionin a vicinity of the origin O = (0, 0), and one can write a segment of the Taylor seriesy = y(0) + xy ′ (0) + x22 y′′ (ηx) = 1 2 x2 y ′′ (ηx) (0 ≤ η < 1)because y(0) = y ′ (0) = 0. Denoting r = |x − y| and taking into account that x = (x, y) andthe origin of coordinates is placed at the point y = (0, 0), we obtainandr =√√√(x − 0) 2 + (y − 0) 2 = x 2 + y 2 = x 2 + 1 √4 x4 (y ′′ (ηx)) 2 = x 1 + 1 4 x2 (y ′′ (ηx)) 2 ;cos θ x,y = y r = 1 xy ′′ (ηx)2√1 + x 2 (1/4)(y ′′ (ηx)) ,2cos θ x,yrThe curvature K of a plane curve is given bywhich yields y ′′ (0) = K(y), and, finally,= y r = 1 y ′′ (ηx)2 2 1 + x 2 (1/4)(y ′′ (ηx)) . 2K =cos θ x,ylimr→0 ry ′′(1 + (y ′ ) 2 ) 3/2 ,= 1 2 K(y)52