Hilbert情形下广义积分Menger曲率的临界点的解析性
On the Analyticity of Critical Points of the Generalized Integral Menger Curvature in the Hilbert Case
摘要：
本文证明了广义积分Menger曲率能$\mathrm{intM}^{(p,2)}$的光滑临界点的解析性，其中$p \in (\tfrac 73, \tfrac 83)$受固定长度约束.这意味着，连同已经熟知的正则性结果，有限能量，临界$C^1$ -曲线$γ: \mathbb{R}/\mathbb{Z} \to \mathbb{R}^n$的广义积分门格尔曲率$\mathrm{intM}^{(p,2)}$受到固定长度的限制，不仅是$C^\infty$，但也分析。我们的方法的灵感来自于解析结果的临界点奥哈拉结能量的基础上柯西的方法的优势和分解的一阶变分。主要的新思想是一个额外的迭代在递归估计的衍生物，以获得足够的差异的秩序的规律性。
英文摘要：
We prove the analyticity of smooth critical points for generalized integral Menger curvature energies $\mathrm{intM}^{(p,2)}$, with $p \in (\tfrac 73, \tfrac 83)$, subject to a fixed length constraint. This implies, together with already well-known regularity results, that finite-energy, critical $C^1$-curves $γ: \mathbb{R}/\mathbb{Z} \to \mathbb{R}^n$ of generalized integral Menger curvature $\mathrm{intM}^{(p,2)}$ subject to a fixed length constraint are not only $C^\infty$ but also analytic. Our approach is inspired by analyticity results on critical points for O'Hara's knot energies based on Cauchy's method of majorants and a decomposition of the first variation. The main new idea is an additional iteration in the recursive estimate of the derivatives to obtain a sufficient difference in the order of regularity.
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