A fractional notion of length and an associated nonlocal curvature

TL;DR

The paper introduces a new fractional length concept for smooth curves, linking it to nonlocal curvature and connecting it to existing fractional perimeter ideas, with convergence to classical length in a limit.

Contribution

It proposes a novel fractional length measure for curves and defines a nonlocal curvature inspired by fractional perimeter concepts.

Findings

Fractional length converges to classical length in a specific limit.

A nonlocal curvature for curves is derived from the fractional length via Euler-Lagrange equations.

The approach parallels the fractional perimeter's use in defining nonlocal mean curvature.

Abstract

Here a new notion of fractional length of a smooth curve, which depends on a parameter $\sigma$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length up to a multiplicative constant. Since a curve that connects two points of minimal length must have zero curvature, the Euler--Lagrange equation associated with the fractional length is used to motivate a nonlocal notion of curvature for a curve. This is analogous to how the fractional perimeter has been used to define a nonlocal mean curvature.