A notion of nonlocal curvature

Nicola Abatangelo, Enrico Valdinoci

Research output: Contribution to journalArticlepeer-review

39 Citations (Scopus)


We consider a nonlocal (or fractional) curvature and we investigate similarities and differences with respect to the classical local case. In particular, we show that the nonlocal mean curvature can be seen as an average of suitable nonlocal directional curvatures and there is a natural asymptotic convergence to the classical case. Nevertheless, different 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.

Original languageEnglish
Pages (from-to)793-815
Number of pages23
JournalNumerical Functional Analysis and Optimization
Issue number7-9
Publication statusPublished - 3 Jul 2014
Externally publishedYes


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