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 ⊆ ℝ2 this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as s → 1 - of the fractional perimeter of a set having locally finite (classical) perimeter.

Fractional Perimeters from a Fractal Perspective / L. Lombardini. - In: ADVANCED NONLINEAR STUDIES. - ISSN 1536-1365. - (2018 Jun 13). [Epub ahead of print]

Fractional Perimeters from a Fractal Perspective

L. Lombardini
2018-06-13

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 ⊆ ℝ2 this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as s → 1 - of the fractional perimeter of a set having locally finite (classical) perimeter.
Settore MAT/05 - Analisi Matematica
13-giu-2018
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/609150
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