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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2506.07752 |
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| _version_ | 1866918050837561344 |
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| author | Dąbrowski, Damian Orponen, Tuomas |
| author_facet | Dąbrowski, Damian Orponen, Tuomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the logarithmic equilibrium measure on curves Dąbrowski, Damian Orponen, Tuomas Classical Analysis and ODEs 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. |
| title | On the logarithmic equilibrium measure on curves |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2506.07752 |