Boundary behaviour of positive harmonic functions on Lipschitz domains

Tom Carroll

Boundary behaviour of positive harmonic functions on Lipschitz domains

Tom Carroll

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

Christopher Bishop (1991) proved an extension to higher dimensions of a result of Bishop, Carleson, Garnett and Jones (1989) concerning the harmonic measure of the intersection of a disk of radius r and a Jordan curve Γ both with respect to the interior domain for Γ and with respect to the exterior domain. Their result provides the upper bound Cr² for the product of these harmonic measures. We apply the extended result to study the boundary behaviour of positive harmonic functions on Lipschitz domains in R^d and obtain results that, in two dimensions, are related to the angular derivative problem.

Original language	English
Pages (from-to)	231-236
Number of pages	6
Journal	Annales Academiae Scientiarum Fennicae Mathematica
Volume	27
Issue number	1
Publication status	Published - 2002

Cite this

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abstract = "Christopher Bishop (1991) proved an extension to higher dimensions of a result of Bishop, Carleson, Garnett and Jones (1989) concerning the harmonic measure of the intersection of a disk of radius r and a Jordan curve Γ both with respect to the interior domain for Γ and with respect to the exterior domain. Their result provides the upper bound Cr2 for the product of these harmonic measures. We apply the extended result to study the boundary behaviour of positive harmonic functions on Lipschitz domains in Rd and obtain results that, in two dimensions, are related to the angular derivative problem.",

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AB - Christopher Bishop (1991) proved an extension to higher dimensions of a result of Bishop, Carleson, Garnett and Jones (1989) concerning the harmonic measure of the intersection of a disk of radius r and a Jordan curve Γ both with respect to the interior domain for Γ and with respect to the exterior domain. Their result provides the upper bound Cr2 for the product of these harmonic measures. We apply the extended result to study the boundary behaviour of positive harmonic functions on Lipschitz domains in Rd and obtain results that, in two dimensions, are related to the angular derivative problem.

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Boundary behaviour of positive harmonic functions on Lipschitz domains

Abstract

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