Asymptotically Neutral Families in $E^3 $
D. T. Piele · SIAM Journal on Mathematical Analysis · 1973
Consider a bounded, open, connected region D in $E^3 $ with connected complement. For a sufficiently smooth Lyapunov boundary surface S, we construct an asymptotically neutral family$\{ y_{n1} ,y_{n2} , \cdots ,y_{nn} \} $, $n = n_j \to \infty $, of points on S which, by definition, have the property that the sum of the potentials due to unit charges placed at $\{ y_{n1} ,y_{n2} , \cdots ,y_{nn} \} $ converge (modulo constants $C_n $) to zero as $n = n_j \to \infty $. Specifically, $\sum olimits_{k = 1}^n {({1 / {\| {x - y_{n,k} } \|}})} + C_n \to 0$ uniformly on every compact subset $K \subset D$. The fields corresponding to asymptotically neutral families tend to zero uniformly on every compact subset $K \subset D$, $\sum olimits_{k = 1}^n { abla [({1 / {\| {x - y_{n,k} } \|}})]} \to 0$. In the course of the construction we examine: (i) the equilibrium distribution $\mu $ on S, $\int _s {{(\mu (y)} / {\| {x - y} \|)}}d\sigma (y) = C$, and how the Hölder continuous differentiability of $\mu $ is related to that of S; (ii) a proof of the strict positivity of $\mu $ using a result of E. Hopf; (iii) an approximation to the integral $\int _s {{(\mu (y)} / {\| {x - y} \|)}}d\sigma (y)$ by a sum of plane integrals each of which is further approximated by a Gauss-type numerical integration rule. The construction of asymptotically neutral families for bounded simply connected regions in $E^2 $ has been done by Korevaar. New techniques are developed in this paper to extend the results to $E^n $, $n \geqq 3$.