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- W4321648696 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 $N notequiv 0 ,(bmod,, 3)$. 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 $N>2$." @default.
- W4321648696 created "2023-02-24" @default.
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- W4321648696 date "2023-02-21" @default.
- W4321648696 modified "2023-09-26" @default.
- W4321648696 title "3D Koch-type crystals" @default.
- W4321648696 doi "https://doi.org/10.48550/arxiv.2302.10628" @default.
- W4321648696 hasPublicationYear "2023" @default.
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