On the analyticity of critical points of the generalized integral Menger curvature in the Hilbert case. (August 2022) - Steenebrügge, Daniel; Vorderobermeier, Nicole
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.