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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.25096 |
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| _version_ | 1866911545291702272 |
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| author | Liu, Junyuan Peng, Shuangjie Zhong, Fulin |
| author_facet | Liu, Junyuan Peng, Shuangjie Zhong, Fulin |
| contents | We prove that for any bounded convex domain $Ω\subset \mathbb{R}^n$, the function \begin{equation*} ψ_Ω(ξ) = \int_{\mathbb{R}^n\setminusΩ} \frac{\mathrm{d}x}{|x-ξ|^{2n}}, \quad ξ\inΩ, \end{equation*} has exactly one critical point. This confirms an conjecture proposed by Clapp, Pistoia and Saldaña in [J. Math. Pures Appl. 205 (2026), 103783]. The proof uses a spherical coordinates representation to write $ψ_Ω$ as an integral of the distance function $ρ(ξ,ω)$. This approach is not limited to $ψ_Ω$. Instead, it provides a general framework for analyzing a broad class of functionals involving the boundary distance. We also examine non-convex domains. In particular, a single annulus exhibits a full circle of critical points, while multiple concentric annuli produce finitely many critical spheres. These examples show that the convexity hypothesis is essential for the uniqueness conclusion. The method developed here for handling spherical integrals involving the distance function is likely to be useful in other geometric and analytic contexts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25096 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the uniqueness of the critical point of $ψ_Ω$ Liu, Junyuan Peng, Shuangjie Zhong, Fulin Analysis of PDEs We prove that for any bounded convex domain $Ω\subset \mathbb{R}^n$, the function \begin{equation*} ψ_Ω(ξ) = \int_{\mathbb{R}^n\setminusΩ} \frac{\mathrm{d}x}{|x-ξ|^{2n}}, \quad ξ\inΩ, \end{equation*} has exactly one critical point. This confirms an conjecture proposed by Clapp, Pistoia and Saldaña in [J. Math. Pures Appl. 205 (2026), 103783]. The proof uses a spherical coordinates representation to write $ψ_Ω$ as an integral of the distance function $ρ(ξ,ω)$. This approach is not limited to $ψ_Ω$. Instead, it provides a general framework for analyzing a broad class of functionals involving the boundary distance. We also examine non-convex domains. In particular, a single annulus exhibits a full circle of critical points, while multiple concentric annuli produce finitely many critical spheres. These examples show that the convexity hypothesis is essential for the uniqueness conclusion. The method developed here for handling spherical integrals involving the distance function is likely to be useful in other geometric and analytic contexts. |
| title | On the uniqueness of the critical point of $ψ_Ω$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.25096 |