Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral

Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Holder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville's Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Holder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points.