# Commentarii Mathematici Helvetici

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**Volume 81, Issue 4, 2006, pp. 809–825**

**DOI: 10.4171/CMH/75**

Published online: 2006-12-31

A weak Kellogg property for quasiminimizers

Anders Björn^{[1]}(1) Linköping University, Sweden

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 Poincaré inequality.

*Keywords: *Doubling measure, weak Kellogg property, metric space, nonlinear, $p$-harmonic, Poincaré inequality, potential theory, quasiharmonic, quasisuperharmonic, quasisuperminimizer, regular

Björn Anders: A weak Kellogg property for quasiminimizers. *Comment. Math. Helv.* 81 (2006), 809-825. doi: 10.4171/CMH/75