Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

TL;DR

This paper develops vector analysis tools for Dirichlet forms on metric measure spaces and applies them to establish existence and uniqueness of solutions for certain quasilinear PDEs and SPDEs, especially on fractals.

Contribution

It introduces a framework for vector analysis in the setting of Dirichlet forms on metric spaces and applies it to solve quasilinear PDEs and SPDEs without requiring locality.

Findings

Established existence and uniqueness of solutions for quasilinear PDEs.

Extended vector analysis tools to non-local Dirichlet forms.

Applied methods to fractal spaces like Sierpinski carpets.

Abstract

Starting with a regular symmetric Dirichlet form on a locally compact separable metric space $X$, our paper studies elements of vector analysis, $L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on $X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets.