No. 2021.01
On the analyticity of critical points of the generalized integral Menger curvature in the Hilbert Case
D. Steenebrügge and N. Vorderobermeier
Subject: Analyticity, knot energy, generalized integral Menger curvature, method of majorants, fractional Leibniz rule, bootstrapping

Abstract

We prove the analyticity of smooth critical points for generalized integral Menger curvature energies $\mathrm{intM}^{(p,2)}$, with $p \in (\frac{7}{3},\frac{8}{3})$, subject to a fixed length constraint. This implies, together with already well-known regularity results, that finite-energy, critical $C^1$-curves $\gamma : \mathbb{R} / \mathbb{Z} \rightarrow \mathbb{R}^n$ 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.

Reference

Oberwolfach Reports

Nonlinear Anal. 221 (2022), Paper No. 112858

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arXiv:2103.07383