Fractional perimeter from a fractal perspective

2016/03/19 by Luca Lombardini, Lombardini, Luca
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals #Theoretical and Computational Physics #math.AP

paper · pdf · doi:10.48550/arxiv.1603.06088

4 figures. arXiv admin note: substantial text overlap with arXiv:1508.06241

arxiv created 2016/03/19 · openalex publication_date 2016/03/19 · arxiv updated 2016/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Following \citeVisintin, we exploit the fractional perimeter of a set to give a definition of fractal dimension for its measure theoretic boundary. 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⊂\mathbb R² this fractal dimension coincides with the Minkowski dimension, namely P_s(S)<∞ \Longleftrightarrow s∈(0,2-(log4)/(log3)). We also study the asymptotics as s→1^- of the fractional perimeter of a set having finite (classical) perimeter.

Citations

• Hitchhikerʼs guide to the fractional Sobolev spaces
• Fractal Geometry: Mathematical Foundations and Applications.
• Fractional Perimeter and Nonlocal Minimal Surfaces

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