Abstract
We study the discretization of a Dirichlet form on the Koch snowflake domain and its boundary with the property that both the interior and the boundary can support positive energy. We compute eigenvalues and eigenfunctions, and demonstrate the localization of high energy eigenfunctions on the boundary via a modification of an argument of Filoche and Mayboroda. Hölder continuity and uniform approximation of eigenfunctions are also discussed.
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Acknowledgements
The authors are grateful to Michael Hinz, Maria Rosaria Lancia, Svitlana Mayboroda, Anna Rozanova-Pierrat, and Tatiana Toro for helpful discussions.
The authors are grateful to Kevin Marinelli for the support in implementing the numerical codes.
Research supported in part by NSF DMS Grants 1659643 and 1613025.
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Gabbard, M., Lima, C., Mograby, G., Rogers, L., Teplyaev, A. (2021). Discretization of the Koch Snowflake Domain with Boundary and Interior Energies. In: Lancia, M.R., Rozanova-Pierrat, A. (eds) Fractals in Engineering: Theoretical Aspects and Numerical Approximations. SEMA SIMAI Springer Series(), vol 8. Springer, Cham. https://doi.org/10.1007/978-3-030-61803-2_4
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