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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$.
@article{10_4007_annals_2010_172_1907,
author = {John Lewis and Kaj Nystr\"om },
title = {Boundary behavior and the {Martin} boundary problem for $p$ harmonic functions in {Lipschitz} domains},
journal = {Annals of mathematics},
pages = {1907--1948},
publisher = {mathdoc},
volume = {172},
number = {3},
year = {2010},
doi = {10.4007/annals.2010.172.1907},
mrnumber = {2726103},
zbl = {1210.31004},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.1907/}
}
TY - JOUR AU - John Lewis AU - Kaj Nyström TI - Boundary behavior and the Martin boundary problem for $p$ harmonic functions in Lipschitz domains JO - Annals of mathematics PY - 2010 SP - 1907 EP - 1948 VL - 172 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.1907/ DO - 10.4007/annals.2010.172.1907 LA - en ID - 10_4007_annals_2010_172_1907 ER -
%0 Journal Article %A John Lewis %A Kaj Nyström %T Boundary behavior and the Martin boundary problem for $p$ harmonic functions in Lipschitz domains %J Annals of mathematics %D 2010 %P 1907-1948 %V 172 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.1907/ %R 10.4007/annals.2010.172.1907 %G en %F 10_4007_annals_2010_172_1907
John Lewis; Kaj Nyström . Boundary behavior and the Martin boundary problem for $p$ harmonic functions in Lipschitz domains. Annals of mathematics, Tome 172 (2010) no. 3, pp. 1907-1948. doi: 10.4007/annals.2010.172.1907
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