Sobolev spaces and calculus of variations on fractals

TL;DR

This paper reviews Sobolev spaces and p-energies on metric measure spaces with Dirichlet forms, extending calculus of variations results to fractals and similar non-classical structures.

Contribution

It generalizes calculus of variations results to fractals and other complex spaces using Sobolev spaces defined via Dirichlet forms.

Findings

Existence of minimizers for convex functionals on fractals.

Extension of calculus of variations to non-classical spaces.

Application to degenerate and superimposed diffusions.

Abstract

The present note contains a review of $p$-energies and Sobolev spaces on metric measure spaces that carry a strongly local regular Dirichlet form. These Sobolev spaces are then used to generalize some basic results from the calculus of variations, such as the existence of minimizers for convex functionals and certain constrained mimimization problems. This applies to a number of non-classical situations such as degenerate diffusions, superpositions of diffusions and diffusions on fractals or on products of fractals.