Vector analysis on fractals and applications

TL;DR

This paper reviews recent advances in vector analysis on fractals, utilizing Dirichlet forms and 1-forms to study complex PDEs on fractal spaces, enabling new analytical approaches.

Contribution

It introduces a framework for vector analysis on fractals using Dirichlet forms and 1-forms, facilitating the study of PDEs previously inaccessible on fractal structures.

Findings

Development of a weak and non-local vector analysis framework

Application to scalar and vector PDEs on fractals

Extension to localized, pointwise vector analysis

Abstract

The paper surveys some recent results concerning vector analysis on fractals. We start with a local regular Dirichlet form and use the framework of 1-forms and derivations introduced by Cipriani and Sauvageot to set up some elements of a related vector analysis in weak and non-local formulation. This allows to study various scalar and vector valued linear and non-linear partial differential equations on fractals that had not been accessible before. Subsequently a stronger (localized, pointwise or fiberwise) version of this vector analysis can be developed, which is related to previous work of Kusuoka, Kigami, Eberle, Strichartz, Hino, Ionescu, Rogers, R\"ockner, and the authors.