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Some types of regularity for the Dirichlet problem

Published online by Cambridge University Press:  22 January 2016

Amar Sadi
Amar Sadi*
Affiliation:
Departement de Mathematiques, Université de Tizi-Ouzou, Algeria
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The question of whether the existence of a harmonic majorant in a relative neighbourhood of each point of a boundary of a domain D implies the existence of a harmonic majorant in the whole of D has received great attention in recent years and has been dealt with by several authors in different settings. The most general results to date have been achieved in [10] with the Martin boundary. In [9], the author arrives, by independent means, at the conclusions of [10] in the particular case where D is a Lipschitz domain.

In this paper, we answer the question in domains with suitably regular topological frontiers. Our methods rely heavily on the possibility of obtaining an extented-representation for nonnegative superharmonic functions defined near a frontier point. This naturally led to the introduction and the study of new types of regularity for the generalised Dirichlet problem. As well as their suitability in dealing with the question of harmonic majorisation, they present an intrinsic importance as natural extensions of the (classical) regularity. For simplicity reasons, we will treat the finite boundary points and the point at infinity separately.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 126 , June 1992 , pp. 103 - 124
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

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