Boundary regularity for quasiminima of double-phase problems on metric spaces

TL;DR

This paper establishes boundary regularity for quasiminima of double-phase functionals in metric spaces, providing conditions for Hölder continuity at boundary points using variational and phase analysis techniques.

Contribution

It introduces a sufficient boundary regularity condition for quasiminima of double-phase functionals in metric measure spaces, extending regularity theory to more general settings.

Findings

Hölder continuity at boundary points under certain conditions

A variational approach using De Giorgi-type conditions

Boundary regularity results in metric measure spaces

Abstract

We give a sufficient condition for H\"older continuity at a boundary point for quasiminima of double-phase functionals of $p,q$-Laplace type, in the setting of metric measure spaces equipped with a doubling measure and supporting a Poincar\'e inequality. We use a variational approach based on De Giorgi-type conditions to give a pointwise estimate near a boundary point. The proofs are based on a careful phase analysis and estimates in the intrinsic geometries.