Fractional perimeter from a fractal perspective

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.