A weak Kellogg property for quasiminimizers

The Kellogg property says that the set of irregular boundary points has capacity zero, i.e. given a bounded open set $\Omega$ there is a set $E \subset \partial\Omega$ with capacity zero such that for all $p$-harmonic functions $u$ in $\Omega$ with continuous boundary values in Sobolev sense, $u$ attains its boundary values at all boundary points in $\partial\Omega \setminus E$. In this paper, we show a weak Kellogg property for quasiminimizers: a quasiminimizer with continuous boundary values in Sobolev sense takes its boundary values at quasievery boundary point. The exceptional set may however depend on the quasiminimizer. To obtain this result we use the potential theory of quasisuperminimizers and prove a weak Kellogg property for quasisuperminimizers. This is done in complete doubling metric spaces supporting a Poincare inequality.