Multivariable Advanced Calculus

330 THE LAPLACE AND POISSON EQUATIONSThis term does not disappear as ε → 0. First note that since ψ x has bounded derivatives,∫ ( ) ( ∫lim − ∂rn ∂ψxu (y) (y − x) −ε→0∂B ε∂n ∂n (y) dσ = lim − u (y) ∂r )nε→0∂B ε∂n (y − x) dσ (12.7)and so it is just this last item which is of concern.First consider the case that n = 2. In this case,( )y 1∇r 2 (y) =|y| 2 , y 2|y| 2Also, on ∂B ε , the exterior unit normal, n, equalsIt follows that on ∂B ε ,1ε (y 1 − x 1 , y 2 − x 2 ) .∂r 2∂n (y − x) = 1 ε (y 1 − x 1 , y 2 − x 2 ) ·Therefore, this term in 12.7 converges to()y 1 − x 1|y − x| 2 , y 2 − x 2|y − x| 2 = 1 ε .Next consider the case where n ≥ 3. In this case,(y1∇r n (y) = − (n − 2)|y| n , · · · , y )n|y|and the unit outer normal, n, equalsTherefore,∂r n∂n− u (x) 2π. (12.8)1ε (y 1 − x 1 , · · · , y n − x n ) .(y − x) = −(n− 2)ε|y − x| 2|y − x| n =− (n − 2)ε n−1 .Letting ω n denote the n−1 dimensional surface area of the unit sphere, S n−1 , it followsthat the last term in 12.7 converges toFinally consider the integral,u (x) (n − 2) ω n (12.9)∫B εv∆udx.∫∫|v∆u| dx ≤ C |r n (y − x) − ψ x (y)| dyB ε B∫ε≤ C |r n (y − x)| dy + O (ε n )B εUsing polar coordinates to evaluate this improper integral in the case where n ≥ 3,∫∫ ε ∫1C |r n (y − x)| dx = CB ε 0 S ρ n−2 ρn−1 dσdρ∫ n−1 ε ∫= C ρdσdρ0 S n−1