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A fractional notion of length and an associated nonlocal curvature
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A fractional notion of length and an associated nonlocal 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.
Forward citations
Cited by 1 Pith paper
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First variation of the fractional $k$-dimensional measure: extending the concept of nonlocal curvature to submanifolds
Derives first variation of fractional k-measure on submanifolds and defines nonlocal mean-curvature vector agreeing with prior hypersurface results.
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