A note on the radially symmetry in the moving plane method
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| Format: | Preprint |
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2024
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| _version_ | 1866929501520265216 |
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| author | Hsu, Shu-Yu |
| author_facet | Hsu, Shu-Yu |
| contents | Let $Ω\subset\mathbb{R}^n$, $n\ge 2$, be a bounded connected $C^2$ domain. For any unit vector $ν\in\mathbb{R}^n$, let $T_λ^ν=\{x\in\mathbb{R}^n:x\cdotν=λ\}$, $Σ_λ^ν=\{x\inΩ:x\cdotν<λ\}$ and $x^{\ast}=x-2(x\cdotν-λ)ν$ be the reflection of a point $x\in\mathbb{R}^n$ about the plane $T_λ^ν$. Let $\widetildeΣ_λ^ν=\{x\inΩ:x^{\ast}\inΣ_λ^ν\}$ and $u\in C^2(\overlineΩ)$. Suppose for any unit vector $ν\in\mathbb{R}^n$, there exists a constant $λ_ν\in\mathbb{R}$ such that $Ω$ is symmetric about the plane $T_{λ_ν}^ν$ and $u$ is symmetric about the plane $T_{λ_ν}^ν$ and satisfies (i)$\,\frac{\partial u}{\partialν}(x)>0\quad\forall x\in Σ_{λ_ν}^ν$ and (ii)$\,\frac{\partial u}{\partialν}(x)<0\quad\forall x\in \widetildeΣ_{λ_ν}^ν$. We will give a simple proof that $u$ is radially symmetric about some point $x_0\inΩ$ and $Ω$ is a ball with center at $x_0$. Similar result holds for the domain $\mathbb{R}^n$ and function $u\in C^2(\mathbb{R}^n)$ satisfying similar monotonicity and symmetry conditions. We also extend this result under weaker hypothesis on the function $u$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_10834 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the radially symmetry in the moving plane method Hsu, Shu-Yu Analysis of PDEs AMS 2020 Mathematics Subject Classification: Primary 35J61 Secondary 35J25 Let $Ω\subset\mathbb{R}^n$, $n\ge 2$, be a bounded connected $C^2$ domain. For any unit vector $ν\in\mathbb{R}^n$, let $T_λ^ν=\{x\in\mathbb{R}^n:x\cdotν=λ\}$, $Σ_λ^ν=\{x\inΩ:x\cdotν<λ\}$ and $x^{\ast}=x-2(x\cdotν-λ)ν$ be the reflection of a point $x\in\mathbb{R}^n$ about the plane $T_λ^ν$. Let $\widetildeΣ_λ^ν=\{x\inΩ:x^{\ast}\inΣ_λ^ν\}$ and $u\in C^2(\overlineΩ)$. Suppose for any unit vector $ν\in\mathbb{R}^n$, there exists a constant $λ_ν\in\mathbb{R}$ such that $Ω$ is symmetric about the plane $T_{λ_ν}^ν$ and $u$ is symmetric about the plane $T_{λ_ν}^ν$ and satisfies (i)$\,\frac{\partial u}{\partialν}(x)>0\quad\forall x\in Σ_{λ_ν}^ν$ and (ii)$\,\frac{\partial u}{\partialν}(x)<0\quad\forall x\in \widetildeΣ_{λ_ν}^ν$. We will give a simple proof that $u$ is radially symmetric about some point $x_0\inΩ$ and $Ω$ is a ball with center at $x_0$. Similar result holds for the domain $\mathbb{R}^n$ and function $u\in C^2(\mathbb{R}^n)$ satisfying similar monotonicity and symmetry conditions. We also extend this result under weaker hypothesis on the function $u$. |
| title | A note on the radially symmetry in the moving plane method |
| topic | Analysis of PDEs AMS 2020 Mathematics Subject Classification: Primary 35J61 Secondary 35J25 |
| url | https://arxiv.org/abs/2409.10834 |