Fractional Perimeters from a Fractal Perspective

Abstract

Abstract The purpose of this paper consists in a better understanding of the fractional nature of the nonlocal perimeters introduced in [L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 2010, 9, 1111–1144]. Following [A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 1991, 2, 175–201], we exploit these fractional perimeters to introduce a definition of fractal dimension for the measure theoretic boundary of a set. We calculate the fractal dimension of sets which can be defined in a recursive way, and we give some examples of this kind of sets, explaining how to construct them starting from well-known self-similar fractals. In particular, we show that in the case of the von Koch snowflake S ⊆ ℝ 2 {S\subseteq\mathbb{R}^{2}} this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as s → 1 - {s\to 1^{-}} of the fractional perimeter of a set having locally finite (classical) perimeter.