Riesz $s$-equilibrium measures on $d$-rectifiable sets as $s$ approaches $d$

Abstract

Let $A$ be a compact set in ${\mathbb R}^p$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^s$ is the unique Borel probability measure with support in $A$ that minimizes $$ I_s(\mu):=\iint\frac{1}{|x-y|^s}d\mu(y)d\mu(x)$$ over all such probability measures. If $A$ is strongly $({\mathcal H}^d, d)$-rectifiable, then $\mu^s$ converges in the weak-star topology to normalized $d$-dimensional Hausdorff measure restricted to $A$ as $s$ approaches $d$ from below.