Published Paper

Inserted: 19 sep 2023
Last Updated: 19 sep 2023

Journal: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Volume: 60
Year: 2021
Doi: https://doi.org/10.1007/s00526-021-02099-y

Abstract:

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.