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Journal article
HARMONIC GRADIENTS ON HIGHER-DIMENSIONAL SIERPITSKI GASKETS
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY, v 28(6), 2050108
Sep 2020
Abstract
We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpitiski Gasket and its higher-dimensional variants SG(N), N > 3, proving results that generalize those of Teplyaev (Gradients on fractals, J. Funct. Anal. 174(1) (2000) 128-1541. When SG(N) is equipped with the standard Dirichlet form and measure mu, we show there is a full mu-measure set on which continuity of the Laplacian implies existence of the gradient del u, and that this set is not all of SG(N). We also show there is a class of non-uniform measures on the usual Sierphiski Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere in sharp contrast to the case with the standard measure.
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Details
- Title
- HARMONIC GRADIENTS ON HIGHER-DIMENSIONAL SIERPITSKI GASKETS
- Publication Details
- FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY, v 28(6), 2050108
- Publisher
- WORLD SCIENTIFIC PUBL CO PTE LTD; SINGAPORE
- Grant note
- This research is supported in part by the NSF Grants DMS-1659643 and DMS-1613025. The authors are grateful to Alexander Teplyaev and Daniel Kelleher for helpful discussions.
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Drexel University
- Web of Science ID
- WOS:000587733300009
- Scopus ID
- 2-s2.0-85092890215
- Other Identifier
- 991021860685304721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics, Interdisciplinary Applications