Nonhomogeneous Variational Problems and Quasi-Minimizers on Metric Spaces

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.