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The Fundamental Function of the Distance

The Fundamental Function of the Distance M. J. PARKER 1. Introduction and main results Throughout this paper D is a domain in the Euclidean space U (N ^ 2). By CD we denote the complement of D in R and by 6Z> the finite boundary of D. For a given x in R and a positive real number r, B(x, r) denotes the open ball centre x, radius r, and S(x, r) denotes the sphere centred at JC and of radius r. For a function /integrable with respect to the surface area measure a on S{xj) the mean of/over S(x, r) is given by I p JH (/ , x, r) = — I fyx •+• r/) u<r(/), ^iv «'s(o, i) where o ^ is the surface area of the unit sphere in U . When D is a proper subdomain of U we define the distance function d:D->(0, +oo) by </(*) = dist (x, CZ)) = dist (JC, dD) = inf || x -1 \\ (x e /)) , <e5£» where || • || denotes the Euclidean norm. We define the function T: [0, + oo) -»M by — log/7 N=2 for all v in (0, + oo), and T(0) = +oo. So for http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

The Fundamental Function of the Distance

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/19.4.337
Publisher site
See Article on Publisher Site

Abstract

M. J. PARKER 1. Introduction and main results Throughout this paper D is a domain in the Euclidean space U (N ^ 2). By CD we denote the complement of D in R and by 6Z> the finite boundary of D. For a given x in R and a positive real number r, B(x, r) denotes the open ball centre x, radius r, and S(x, r) denotes the sphere centred at JC and of radius r. For a function /integrable with respect to the surface area measure a on S{xj) the mean of/over S(x, r) is given by I p JH (/ , x, r) = — I fyx •+• r/) u<r(/), ^iv «'s(o, i) where o ^ is the surface area of the unit sphere in U . When D is a proper subdomain of U we define the distance function d:D->(0, +oo) by </(*) = dist (x, CZ)) = dist (JC, dD) = inf || x -1 \\ (x e /)) , <e5£» where || • || denotes the Euclidean norm. We define the function T: [0, + oo) -»M by — log/7 N=2 for all v in (0, + oo), and T(0) = +oo. So for

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jul 1, 1987

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