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

Abstract

In this paper boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points 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 complete metric spaces equipped with doubling measures supporting a p-Poincaré inequality, but the results are new also in unweighted Euclidean spaces.Note that by Proposition 3.5, C p (∂ M Ω, Ω M ) > 0, so the Kellogg property is never trivial. The following uniqueness result is also new. Theorem 1.3. Assume that Ω is a bounded domain which is finitely connected at the boundary. Let f ∈ C(∂ M Ω). Then there exists a unique bounded p-harmonic function u on Ω such thatMoreover, u equals the Perron solution P Ω M f .We are also able to show that boundary regularity is a local property for the Mazurkiewicz boundary in the following sense.Theorem 1.4. Assume that Ω is a bounded domain which is finitely connected at the boundary. Letx ∈ ∂ M Ω and let G be an Ω M -neighbourhood ofx.Thenx is regular with respect to Ω M if and only if it is regular with respect toThroughout the paper we also study to what extent such results are true for other compactifications of Ω. The details describing the different results and cases are quite involved.Boundary regularity for p-harmonic functions with respect to the given metric boundary has been studied for a long period, especially on R n . The first significant result was Maz ′ ya's [37] sufficiency part of the Wiener criterion in 1970. Later on the full Wiener criterion was obtained in various situations including weighted R n and for Cheeger p-harmonic functions on metric spaces, see [25], [32], [35], [39] and [18]. The full Wiener criterion remains open (for the given metric boundary) in the generality considered here, but the sufficiency has been obtained, see [21] and [19], and a weaker necessity condition, see [20].Abstract. We fill in a gap in the proofs of Theorems 1.1-1.4 in "The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications", to appear in

Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular spaces

“…Boundary regularity with respect to the Mazurkiewicz boundary in bounded domains that are finitely connected at the boundary was studied in Björn [6]. The results therein can therefore be reformulated using sphericalization for unbounded domains as well.…”