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 Poincar'{e} inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally H"{o}lder 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 H"older 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.