Fractional perimeter from a fractal perspective

TL;DR

This paper explores the fractional perimeter as a tool to define and analyze fractal dimensions of measure-theoretic boundaries, with applications to self-similar fractals like the von Koch snowflake, linking fractional perimeter to Minkowski dimension.

Contribution

It introduces a fractal dimension based on fractional perimeter and demonstrates its equivalence to Minkowski dimension for certain fractals, including the von Koch snowflake.

Findings

Fractional perimeter characterizes fractal dimensions of measure-theoretic boundaries.

For the von Koch snowflake, the fractional perimeter's finiteness relates to a specific range of s.

As s approaches 1, the fractional perimeter's asymptotics are studied for sets with finite classical perimeter.

Abstract

Following \cite{Visintin}, 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 $S\subset\mathbb R^2$ this fractal dimension coincides with the Minkowski dimension, namely \begin{equation*} P_s(S)<\infty\qquad\Longleftrightarrow\qquad s\in\Big(0,2-\frac{\log4}{\log3}\Big). \end{equation*} We also study the asymptotics as $s\to1^-$ of the fractional perimeter of a set having finite (classical) perimeter.