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Title: HARMONIC GRADIENTS ON HIGHER-DIMENSIONAL SIERPIŃSKI GASKETS
We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpiński Gasket and its higher-dimensional variants [Formula: see text], [Formula: see text], proving results that generalize those of Teplyaev [Gradients on fractals, J. Funct. Anal. 174(1) (2000) 128–154]. When [Formula: see text] is equipped with the standard Dirichlet form and measure [Formula: see text] we show there is a full [Formula: see text]-measure set on which continuity of the Laplacian implies existence of the gradient [Formula: see text], and that this set is not all of [Formula: see text]. We also show there is a class of non-uniform measures on the usual Sierpiński 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.  more » « less
Award ID(s):
1659643
PAR ID:
10211256
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
Fractals
Volume:
28
Issue:
06
ISSN:
0218-348X
Page Range / eLocation ID:
2050108
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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