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

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