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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.08881 |
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| _version_ | 1866913234323243008 |
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| author | Kenig, Carlos Zhao, Zihui |
| author_facet | Kenig, Carlos Zhao, Zihui |
| contents | Let $u$ be a harmonic function in a $C^1$ domain $D\subset \mathbb{R}^d$, which vanishes on an open subset of the boundary. In this note we study its critical set $\{x \in \overline{D}: \nabla u(x) = 0 \}$. When $D$ is a $C^{1,α}$ domain for some $α\in (0,1]$, we give an upper bound on the $(d-2)$-dimensional Hausdorff measure of the critical set by the frequency function. We also discuss possible ways to extend such estimate to all $C^1$-Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set $\{x \in \overline{D}: u(x) = 0 = |\nabla u(x)| \}$ (see [KZ1, KZ2]). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08881 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the critical set of harmonic functions near the boundary Kenig, Carlos Zhao, Zihui Analysis of PDEs Let $u$ be a harmonic function in a $C^1$ domain $D\subset \mathbb{R}^d$, which vanishes on an open subset of the boundary. In this note we study its critical set $\{x \in \overline{D}: \nabla u(x) = 0 \}$. When $D$ is a $C^{1,α}$ domain for some $α\in (0,1]$, we give an upper bound on the $(d-2)$-dimensional Hausdorff measure of the critical set by the frequency function. We also discuss possible ways to extend such estimate to all $C^1$-Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set $\{x \in \overline{D}: u(x) = 0 = |\nabla u(x)| \}$ (see [KZ1, KZ2]). |
| title | A note on the critical set of harmonic functions near the boundary |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.08881 |