TY - JOUR
T1 - Hausdorff dimension and σ finiteness of p harmonic measures in space when p ≥ n
AU - Akman, Murat
AU - Lewis, John
AU - Vogel, Andrew
N1 - Funding Information:
Akman and Lewis were partially supported by NSF DMS-0900291 .
Publisher Copyright:
© 2015 Elsevier Ltd. All rights reserved.
PY - 2015/12/1
Y1 - 2015/12/1
N2 - In this paper we study a measure, μ associated with a positive p harmonic function û defined in an open set O⊂ℝRn and vanishing on a portion Γ of ∂O. If p>n we show μ is concentrated on a set of σ finite Hn-1 measure while if p=n the same conclusion holds provided Γ is uniformly fat in the sense of n capacity. Our work nearly answers in the affirmative a conjecture in Lewis (2015) and also appears to be the natural extension of Jones and Wolff (1988), Wolff (1993), to higher dimensions.
AB - In this paper we study a measure, μ associated with a positive p harmonic function û defined in an open set O⊂ℝRn and vanishing on a portion Γ of ∂O. If p>n we show μ is concentrated on a set of σ finite Hn-1 measure while if p=n the same conclusion holds provided Γ is uniformly fat in the sense of n capacity. Our work nearly answers in the affirmative a conjecture in Lewis (2015) and also appears to be the natural extension of Jones and Wolff (1988), Wolff (1993), to higher dimensions.
KW - MSC primary 35J25
KW - secondary 35J70
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U2 - 10.1016/j.na.2015.08.021
DO - 10.1016/j.na.2015.08.021
M3 - Article
AN - SCOPUS:84942423193
SN - 0362-546X
VL - 129
SP - 198
EP - 216
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
ER -