# Boundary behavior and the Martin boundary problem for $p$ harmonic functions in Lipschitz domains

### Abstract

In a previous article, we proved a boundary Harnack inequality for the ratio of two positive $p$ harmonic functions, vanishing on a portion of the boundary of a Lipschitz domain. In the current paper we continue our study by showing that this ratio is Hölder continuous up to the boundary. We also consider the Martin boundary of certain domains and the corresponding question of when a minimal positive $p$ harmonic function (with respect to a given boundary point $w$) is unique up to constant multiples. In particular we show that the Martin boundary can be identified with the topological boundary in domains that are convex or $C^1$. Minimal positive $p$ harmonic functions relative to a boundary point $w$ in a Lipschitz domain are shown to be unique, up to constant multiples, provided the boundary is sufficiently flat at $w$.

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## Authors

John Lewis

Department of Mathematics
University of Kentucky
Lexington, KY 40506-0027
United States

Kaj Nyström

Department of Mathematics
Umeå University
S-90187 Umeå
Sweden