On the logarithmic equilibrium measure on curves

Abstract

Let $μ$ be the logarithmic equilibrium measure on a compact set $γ\subset \mathbb{R}^{d}$. We prove that $μ$ is absolutely continuous with respect to the length measure on the part of $γ$ which can be locally expressed as the graph of a $C^{1,α}$-function $\mathbb{R} \to \mathbb{R}^{d - 1}$, $α> 0$. For $d = 2$, at least in the case where $γ$ is a compact $C^{1,α}$-graph, our result can also be deduced from the classical fact that $μ$ coincides with the harmonic measure of $Ω=\mathbb{R}^{2} \, \setminus \, γ$ with pole at $\infty$. For $d \geq 3$, however, our result is new even for $C^{\infty}$-graphs. In fact, up to now it was not even known if the support of $μ$ has positive dimension.