Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 149-181

TOWARDS A REGULARITY THEORY FOR INTEGRAL MENGER CURVATURE

Simon Blatt and Philipp Reiter

Karlsruhe Institute of Technology, Institute for Analysis
Kaiserstrasse 12, 76131 Karlsruhe, Germany; simon.blatt 'at' kit.edu

Universität Duisburg-Essen, Fakultät für Mathematik
Forsthausweg 2, 47057 Duisburg, Germany; philipp.reiter 'at' uni-due.de

Abstract. We generalize the notion of integral Menger curvature introduced by Gonzalez and Maddocks [14] by decoupling the powers in the integrand. This leads to a new two-parameter family of knot energies intM^(p,q). We classify finite-energy curves in terms of Sobolev–Slobodeckii spaces. Moreover, restricting to the range of parameters leading to a sub-critical Euler–Lagrange equation, we prove existence of minimizers within any knot class via a uniform bi-Lipschitz bound. Consequently, intM^(p,q) is a knot energy in the sense of O'Hara. Restricting to the non-degenerate sub-critical case, a suitable decomposition of the first variation allows to establish a bootstrapping argument that leads to C^∞-smoothness of critical points.

2010 Mathematics Subject Classification: Primary 42A45, 53A04, 57M25.

Key words: Knot energy, Menger curvature, integral Menger curvature, regularity, fractional Sobolev spaces, fractional seminorms.

Reference to this article: S. Blatt and P. Reiter: Towards a regularity theory for integral Menger curvature. Ann. Acad. Sci. Fenn. Math. 40 (2015), 149-181.

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doi:10.5186/aasfm.2015.4006