On the Analyticity of Critical Points of the Generalized Integral Menger Curvature in the Hilbert Case

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 $\gamma :\mathbb{R}/\mathbb{Z}\to {\mathbb{R}}^n$ 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.

31 pages

References in corpus (2)

• Gevrey regularity for integro-differential operators
• Analyticity for Solution of Integro-Differential Operators