Nonhomogeneous Variational Problems and Quasi-Minimizers on Metric Spaces

TL;DR

This paper proves that quasi-minimizers of certain energy functionals on metric measure spaces are locally H"older continuous and satisfy the Harnack inequality, under doubling and Poincaré conditions, using De Giorgi and expansion of positivity methods.

Contribution

It extends regularity results for quasi-minimizers to non-homogeneous functionals on metric spaces, combining De Giorgi techniques with new adaptations.

Findings

Quasi-minimizers are locally H"older continuous.

Quasi-minimizers satisfy the Harnack inequality.

Results apply to metric measure spaces with doubling and Poincaré conditions.

Abstract

We show that quasi-minimizers of non-homogeneous energy functionals on metric measure spaces are locally H\"older continuous and satisfy the Harnack inequality. We assume that the spaces are doubling and support a Poincar\'e inequality. The proof is based on the De Giorgi method, combined with the "expansion of positivity" technique.