Using the theory of Sobolev spaces on a metric measure space we are able to apply calculus of variations and define p-harmonic functions as minimizers of the p-Dirichlet integral. More generally, we study regularity properties of quasi-minimizers of p-Dirichlet integrals in a metric measure space. Applying the De Giorgi method we show that quasiminimizers, and in particular p-harmonic functions, satisfy Harnack's inequality, the strong maximum principle, and are locally Hölder continuous, if the space is doubling and supports a Poincaré inequality.

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