- 12-153 Nicola Abatangelo, Enrico Valdinoci
- A notion of nonlocal curvature
Dec 20, 12
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Abstract. We consider a nonlocal (or fractional) curvature
and we investigate similarities and differences with respect to the classical
case. In particular, we show that
the nonlocal mean curvature may be seen as an average of suitable
nonlocal directional curvatures and there is a natural asymptotic convergence to the classical case.
Nevertheless, differently from the classical cases, minimal and maximal nonlocal directional curvatures
are not in general attained at perpendicular directions and in fact one can
arbitrarily prescribe the set of extremal directions for nonlocal directional curvatures.
Also the classical directional
curvatures naturally enjoy some linear
properties that are lost
in the nonlocal framework.
In this sense, nonlocal directional
curvatures are somewhat intrinsically nonlinear.