3D Koch-type crystals

2023/02/21 by Ferrer, Giovanni, Vélez-Santiago, Alejandro · 2 citations
#2020 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2302.10628

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" \C_N\, where N>1 are integers with N \not≡ 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(N²+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 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 C_N. As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals C_N, for N>2.

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