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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.21214 |
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| _version_ | 1866912558730969088 |
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| author | Foster, Benjamin Gallegos, Josep |
| author_facet | Foster, Benjamin Gallegos, Josep |
| contents | Let $u$ be a harmonic function in the unit ball $B_1 \subset \mathbb R^n$, normalized so that its gradient has magnitude at most 1 on the unit ball. We show that if the gradient of $u$ is $ε$-small in size on a set $E\subset B_{1/2}$ with positive $(n-2+δ)$-dimensional Hausdorff content for some $δ>0$, then $\sup_{B_{1/2}} |\nabla u| \leq C ε^α$ with $C,α>0$ depending only on $n,δ$ and the $(n-2+δ)$-Hausdorff content of $E$. This is an improvement over a similar result of Logunov and Malinnikova that required $δ>1-c_n$ for a small dimensional constant $c_n$ and reaches the sharp threshold for the dimension of the smallness sets from which propagation of smallness can occur. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21214 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Propagation of smallness near codimension two for gradients of harmonic functions Foster, Benjamin Gallegos, Josep Analysis of PDEs 31B05 Let $u$ be a harmonic function in the unit ball $B_1 \subset \mathbb R^n$, normalized so that its gradient has magnitude at most 1 on the unit ball. We show that if the gradient of $u$ is $ε$-small in size on a set $E\subset B_{1/2}$ with positive $(n-2+δ)$-dimensional Hausdorff content for some $δ>0$, then $\sup_{B_{1/2}} |\nabla u| \leq C ε^α$ with $C,α>0$ depending only on $n,δ$ and the $(n-2+δ)$-Hausdorff content of $E$. This is an improvement over a similar result of Logunov and Malinnikova that required $δ>1-c_n$ for a small dimensional constant $c_n$ and reaches the sharp threshold for the dimension of the smallness sets from which propagation of smallness can occur. |
| title | Propagation of smallness near codimension two for gradients of harmonic functions |
| topic | Analysis of PDEs 31B05 |
| url | https://arxiv.org/abs/2508.21214 |