A Notion of Nonlocal Curvature

Abstract

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.

Keywords:

• Nonlocal directional curvature
• Nonlocal mean curvature
• Fractional perimeters
• s-minimal surfaces
• Fractional Laplacian

Mathematics Subject Classification:

• 35R11
• 49Q05

ACKNOWLEDGMENTS

It is a pleasure to thank Serena Dipierro for extensive and valuable discussions.

Part of special issue, “Variational Analysis and Applications.”

Notes

Notice that π(e) is simply the portion of the two-dimensional plane spanned by e and ν given by the vectors with positive scalar product with respect to e. We point out that a change of the orientation of ν does not change π(e) which is therefore uniquely defined. Needless to say, such two-dimensional plane plays an important role even in the classical setting, see Figure 1.

We refer here to sets as in footnote 1 in [6].

In further detail, recalling (4),

that is uniformly bounded as s↗1/2. This means that we can suppose that ∂E is a graph in, say, B ₁ and replace it outside B ₁ without affecting the statement of Theorem 12.