Boundary continuity for quasiminimizers on metric spaces

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A weak Kellogg property for quasiminimizers (2006, background): “Björn [10] in metric spaces, they both obtained sufficient conditions for a boundary point to be regular for quasiharmonic functions.”
Yau's gradient estimates on Alexandrov spaces (2010, background): “15 in [5]) proved that ur is Hölder continuous onBp(r).”
Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology (2012, background): “Björn [11], but is slightly simpler and sufficient for our purpose.”
The Perron method for p-harmonic functions in metric spaces (2003, background): “The following result was proved in Björn–Björn–Shanmugalingam [3], Theorem 3.9.”