一个尖锐的平面分数阶等周不等式
A Sharp Planar Fractional Isoperimetric Inequality
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中文总结 AI 辅助
本文证明了平面上分数阶等周不等式的尖锐形式，常数最优且由圆盘取得等号，证明通过约化弦泛函、凸体及分数阶Willmore型不等式等三步完成。
中文摘要 AI 辅助
在本文中，我们证明了平面上分数阶等周不等式的尖锐形式，该问题最初由Maz'ya提出[问题1，积分方程与算子理论，2018]。更精确地，我们证明对于任意给定的$s\in(0,1)$和任意具有$C^1$边界的有界域$\Omega \subset \mathbb{R}^2$，有$$ P_s(\Omega) \le \frac{\pi^{s-\frac{1}{2}}\Gamma\left(\frac{3-s}{2}\right)} {s(1-s)\Gamma\left(\frac{4-s}{2}\right)}\left[\mathcal{H}^1(\partial\Omega)\right]^{2-s}, $$其中$P_s$表示分数阶$s$-周长，$\Gamma$表示Gamma函数，$\mathcal{H}^{1}$表示$\mathbb{R}^2$上的1维Hausdorff测度。该常数是尖锐的，且等号由圆盘取得。证明分三步进行：将分数阶周长约化为弦泛函，将连通域约化为凸体，以及通过分数阶Willmore型不等式、沿外平行体的变分公式和渐近分析证明尖锐的凸弦不等式。
英文摘要
In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given $s\in(0,1)$ and any bounded domain $Ω\subset \mathbb{R}^2$ with $C^1$ boundary, $$ P_s(Ω) \le \frac{π^{s-\frac{1}{2}}Γ\left(\frac{3-s}{2}\right)} {s(1-s)Γ\left(\frac{4-s}{2}\right)}\left[\mathcal{H}^1(\partialΩ)\right]^{2-s}, $$ where $P_s$ denotes the fractional $s$-perimeter, $Γ$ denotes the Gamma function, and $\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure on $\mathbb{R}^2$. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.
