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

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.

References in corpus (4)

• Dirac and magnetic Schrödinger operators on fractals
• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• Martingale dimensions for fractals
• Metrics and spectral triples for Dirichlet and resistance forms

Cited by in corpus (5)

• Dirac and magnetic Schrödinger operators on fractals
• Derivations and Dirichlet forms on fractals
• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• A Feynman-Kac-Itô Formula for magnetic Schrödinger operators on graphs
• Energy measure closability for Dirichlet forms