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

TL;DR

The paper studies the behavior of Riesz $s$-equilibrium measures on $d$-rectifiable sets as the parameter $s$ approaches the set's Hausdorff dimension $d$, showing convergence to the Hausdorff measure.

Contribution

It establishes the weak-star convergence of Riesz $s$-equilibrium measures to the Hausdorff measure on strongly $({ m Hausdorff}^d, d)$-rectifiable sets as $s$ approaches $d$ from below.

Findings

$oldsymbol{ ext{Riesz }s ext{-equilibrium measures converge weak-star to Hausdorff measure}}$

$oldsymbol{ ext{Convergence holds for strongly }({ m Hausdorff}^d, d) ext{-rectifiable sets}$

$oldsymbol{ ext{Results connect potential theory with geometric measure theory}}$

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.