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

Antonella Nastasi, Cintia Pacchiano Camacho

Published 2021-04-13Version 1

Using a variational approach we study regularity properties for quasiminimizers of a $(p,q)$-Dirichlet integral inside a bounded open set, 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\"older continuous and they satisfy Harnack inequality, the strong maximum principle, and Liouville's Theorem. Furthermore, we give a pointwise estimate near a boundary point, also a sufficient condition for H\"older continuity, and a Wiener type regularity condition for continuity of quasiminimizers at a boundary point. Finally, we consider $(p,q)$-minimizers and we give an estimate for their oscillation at boundary points.