12-153 Nicola Abatangelo, Enrico Valdinoci

A notion of nonlocal curvature (339K, pdf) 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.

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