Weak elastic energy of irregular curves

TL;DR

This paper introduces a new weak elastic energy concept for irregular curves in any dimension, based on a relaxation process involving a geometric discrete curvature approximation, linking energy finiteness to second order summability.

Contribution

It proposes a novel $p$-energy for irregular curves using a geometric discrete curvature approach, extending classical curvature integrals to non-regular curves.

Findings

Energy finite iff second order summability of arc-length parameterization

Energy matches the integral of the $p$-th power of scalar curvature for regular curves

Comparison with other discrete curvature definitions included

Abstract

A weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our $p$-energy is defined through a relaxation process, where a suitable $p$-rotation of inscribed polygonals is adopted. The discrete $p$-rotation we choose has a geometric flavor: a polygonal is viewed as an approximation to a smooth curve and hence its discrete curvature is spread out into a smooth density. For any exponent $p$ greater than one, the $p$-energy is finite if and only if the arc-length parameterization of the curve has a second order summability with the same growth exponent. In that case, moreover, the energy agrees with the natural extension of the integral of the $p$-th power of the scalar curvature. Finally, a comparison with other definitions of discrete curvatures is discussed.