Curvature Estimate of Nodal Sets of Harmonic Functions in the Plane
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910589293428736 |
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| author | Sun, Jin |
| author_facet | Sun, Jin |
| contents | In this paper, we study curvature estimates for nodal sets of harmonic functions in the plane. We prove that at any point $p$, the curvature of each nodal curve of any harmonic function $u$ is upper bounded by $$\left|{κ(u)(p)}\right|\leq \frac{4(n+1)}{nr}\cos nα_0,$$ where $u$ has only $n$ nodal curves in $B_r(p)$ intersecting at $p$, and $α_0=0$ for odd $n$ or $α_0=\fracπ{2n(n+1)}$ for even $n$. This result is sharp for all $n\geq 1$. In extreme cases, $u$ can be given by the Poisson extension of Dirac measure and its derivatives. Moreover, the curvature of any nodal curve is uniformly upper bounded at every point in the nodal set of $u$ in a small neighborhood $B_{cr}(p)$, where $c<1$ depends only on $n$. Furthermore, with the frequency tool, we prove that the area of the positive part and the negative part of $u$ have a uniform lower bound, which depends only on the number of nodal domains in $B_r(p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_09022 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curvature Estimate of Nodal Sets of Harmonic Functions in the Plane Sun, Jin Analysis of PDEs In this paper, we study curvature estimates for nodal sets of harmonic functions in the plane. We prove that at any point $p$, the curvature of each nodal curve of any harmonic function $u$ is upper bounded by $$\left|{κ(u)(p)}\right|\leq \frac{4(n+1)}{nr}\cos nα_0,$$ where $u$ has only $n$ nodal curves in $B_r(p)$ intersecting at $p$, and $α_0=0$ for odd $n$ or $α_0=\fracπ{2n(n+1)}$ for even $n$. This result is sharp for all $n\geq 1$. In extreme cases, $u$ can be given by the Poisson extension of Dirac measure and its derivatives. Moreover, the curvature of any nodal curve is uniformly upper bounded at every point in the nodal set of $u$ in a small neighborhood $B_{cr}(p)$, where $c<1$ depends only on $n$. Furthermore, with the frequency tool, we prove that the area of the positive part and the negative part of $u$ have a uniform lower bound, which depends only on the number of nodal domains in $B_r(p)$. |
| title | Curvature Estimate of Nodal Sets of Harmonic Functions in the Plane |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2304.09022 |