Dirichlet forms on metric measure spaces as Mosco limits of Korevaar-Schoen energies

TL;DR

This paper investigates the convergence of Korevaar-Schoen energies to Dirichlet forms on metric measure spaces, providing conditions for Mosco limits and introducing a new Rellich-Kondrachov theorem for these spaces.

Contribution

It offers general conditions for Mosco convergence of Korevaar-Schoen energies and introduces a novel Rellich-Kondrachov theorem for Korevaar-Schoen-Sobolev spaces.

Findings

Established conditions for Mosco limits of Korevaar-Schoen energies

Proved a new Rellich-Kondrachov theorem for Korevaar-Schoen spaces

Extended analysis to Cheeger and fractal-like spaces

Abstract

This paper establishes sufficient general conditions for the existence of Mosco limits of Korevaar-Schoen $L^2$ energies, first in the context of Cheeger spaces and then in the context of fractal-like spaces with walk dimension greater than 2. Among the ingredients, a new Rellich-Kondrachov type theorem for Korevaar-Schoen-Sobolev spaces is of independent interest.