3D Koch-type crystals

Abstract

We consider the construction of a family $\{K_N\}$ of $3$-dimensional Koch-type surfaces, with a corresponding family of $3$-dimensional Koch-type “snowflake analogues” $\{{\mathcal{C}}_N\}$, where $N>1$ are integers with $N\not\equiv 0\mathrm{mod}3$. We first establish that the Koch surfaces $K_N$ are $s_N$-sets with respect to the $s_N$-dimensional Hausdorff measure, for $s_N=\log (N^2+2)/\log (N)$ the Hausdorff dimension of each Koch-type surface $K_N$. Using self-similarity, one deduces that the same result holds for each Koch-type crystal ${\mathcal{C}}_N$. We then develop lower and upper approximation monotonic sequences converging to the $s_N$-dimensional Hausdorff measure on each Koch-type surface $K_N$, and consequently, one obtains upper and lower bounds for the Hausdorff measure for each set ${\mathcal{C}}_N$. As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals ${\mathcal{C}}_N$, for $N>2$.