Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy

Let $\omega$ denote the classical equilibrium distribution (of total charge $1$) on a convex or $C^{1,\alpha }$-smooth conductor $K$ in $\mathbb{R}^{q}$ with nonempty interior. Also, let $\omega _{N}$ be any $N$th order ``Fekete equilibrium distribution'' on $K$, defined by $N$ point charges $1/N$ at $N$th order ``Fekete points''. (By definition such a distribution minimizes the energy for $N$-tuples of point charges $1/N$ on $K$.) We measure the approximation to $\omega$ by $\omega _{N}$ for $N \to \infty$ by estimating the differences in potentials and fields,

both inside and outside the conductor $K$. For dimension $q \geq 3$ we obtain uniform estimates ${\mathcal{O}}(1/N^{1/(q-1)})$ at distance $\geq \varepsilon >0$ from the outer boundary $\Sigma$ of $K$. Observe that ${\mathcal{E}}^{\omega }=0$ throughout the interior $\Omega$ of $\Sigma$ (Faraday cage phenomenon of electrostatics), hence ${\mathcal{E}}^{\omega _{N}}={\mathcal{O}}(1/N^{1/(q-1)})$ on the compact subsets of $\Omega$. For the exterior $\Omega ^{\infty }$ of $\Sigma$ the precise results are obtained by comparison of potentials and energies. Admissible sets $K$ have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries $\Sigma$ and bounds on normal derivatives of interior and exterior Green functions. Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions $\mu _{N}$ of equal point charges on $\Sigma$. In that context there is an important open problem for the sphere which is discussed at the end of the paper.