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arXiv · 1105.1995

Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket

Fabio Cipriani·Daniele Guido·Tommaso Isola·Jean-Luc Sauvageot
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Abstract

We provide a definition of integral, along paths in the Sierpinski gasket K, for differential smooth 1-forms associated to the standard Dirichlet form K. We show how this tool can be used to study the potential theory on K. In particular, we prove: i) a de Rham reconstruction of a 1-form from its periods around lacunas in K; ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; iii) the existence of potentials of smooth 1-forms on a suitable covering space of K. We finally show that this framework provides versions of the de Rham duality theorem for the fractal K.

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Fabio Cipriani, Daniele Guido, Tommaso Isola, Jean-Luc Sauvageot. 2013-01-26. Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket. https://doi.org/10.1016/j.aim.2013.02.014

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