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.