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

TL;DR

This paper investigates the regularity of quasiminimizers of a $(p,q)$-Dirichlet integral in metric spaces, establishing interior and boundary regularity results, including Hölder continuity, Harnack inequality, and boundary estimates.

Contribution

It provides new regularity results for quasiminimizers of $(p,q)$-Dirichlet integrals in metric spaces, extending classical PDE theory to more general settings.

Findings

Quasiminimizers are locally Hölder continuous.

They satisfy Harnack inequality and strong maximum principle.

Boundary regularity conditions are established.

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.