Integral Equations

Since on Ω(x, r) we havegrad y Φ(x, y) = n y2πr , (256)a straightforward calculation using the mean-value theorem and the fact thatfor a harmonic in D function v shows that∫ (∂Φ(x, y)lim u(y)r→0∂n ywhich yields (253).Ω(x,r)12.2 Properties of harmonic functions∫Γ∂v∂n ydl y = 0 (257)− Φ(x, y) ∂u(y) )dl y = u(x), (258)∂n yTheorem 30 Let a twice continuously differentiable function v be harmonic in a domain Dbounded by the closed smooth contour Γ. Then∫Γ∂v(y)∂n ydl y = 0. (259)Proof follows from the first Green’s formula applied to two harmonic functions v and u = 1.Theorem 31 (Mean Value Formula) Let a twice continuously differentiable function u beharmonic in a ball√B(x, r) = {y : |x − y| = (x − x 0 ) 2 + (y − y 0 ) 2 ≤ r}of radius r (a vicinity of a point x) with the boundary Ω(x, r) = {y : |x − y| = r} (a circle)and continuous in the closure ¯B(x, r). Thenu(x) = 1πr 2∫B(x,r)u(y)dy = 12πr∫Ω(x,r)u(y)dl y , (260)i.e., the value of u in the center of the ball is equal to the integral mean values over both theball and its boundary.Proof. For each 0 < ρ < r we apply (259) and Green’s third formula to obtainu(x) = 12πρ∫|x−y|=ρu(y)dl y , (261)whence the second mean value formula in (260) follows by passing to the limit ρ → r. Multiplying(261) by ρ and integrating with respect to ρ from 0 to r we obtain the first mean valueformula in (260).49