On the analyticity of critical points of the generalized integral Menger curvature in the Hilbert case. (August 2022)

• Record Type: Journal Article
• Authors: Steenebrügge, Daniel; Vorderobermeier, Nicole
• Is Part Of: Nonlinear analysis. Volume 221(2022)
• Publication Date: 2022-08
• Subjects: 35A20 -- 35A10 -- 35B65 -- 57K10; Analyticity -- Knot energy -- Generalized integral Menger curvature -- Method of majorants -- Fractional Leibniz rule -- Bootstrapping; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112858
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We prove the analyticity of smooth critical points for generalized integral Menger curvature energies intM ( p, 2 ), with p ∈ ( 7 3, 8 3 ), subject to a fixed length constraint. This implies, together with already well-known regularity results, that finite-energy, critical C 1 -curves γ : R / Z → R n of generalized integral Menger curvature intM ( p, 2 ) subject to a fixed length constraint are not only C ∞ 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.