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In this paper boundary regularity for p-harmonic functions is studied withrespect to the Mazurkiewicz boundary and other compactifications. Inparticular, the Kellogg property (which says that the set of irregular boundarypoints has capacity zero) is obtained for a large class of compactifications,but also two examples when it fails are given. This study is done for completemetric spaces equipped with doubling measures supporting a p-Poincar\'einequality, but the results are new also in unweighted Euclidean spaces.

The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications