Nonhomogeneous Variational Problems and Quasi-Minimizers on Metric Spaces

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