JournalscmhVol. 81 , No. 4DOI 10.4171/cmh/75

A weak Kellogg property for quasiminimizers

  • Anders Björn

    Linköping University, Sweden
A weak Kellogg property for quasiminimizers cover

Abstract

The Kellogg property says that the set of irregular boundary points has capacity zero, i.e. given a bounded open set Ω\Omega there is a set EΩE \subset \partial\Omega with capacity zero such that for all pp-harmonic functions uu in Ω\Omega with continuous boundary values in Sobolev sense, uu attains its boundary values at all boundary points in ΩE\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.