3D Koch-type crystals

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 $N \not\equiv 0 \,(\bmod\,\, 3)$. 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 $N>2$.