Q919495
type: Item
instance of: scholarly article
title: Frontière de Martin d'un domaine de \({\mathbb{R}}^ n\) dont le bord est inclus dans une hypersurface lipschitzienne. (Martin boundary of a dommain in \({\mathbb{R}}^ n\) whose boundary is contained in a Lipschitzian hypersurface)
zbMATH Open document ID: 0707.31012
DOI: 10.1007/BF02386358
author: Nicolas Chevallier
published in: Arkiv för Matematik
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Omega \subset {\mathbb{R}}^ n\) denote an open set such that \(\complement \Omega \equiv F\) is contained in a Lipschitzian hypersurface. For a point \(M\in F\), let \(P_{\Omega}(M)\) denote the cone of positive harmonic functions in \(\Omega\) that vanish on \(F\setminus \{M\}\). It is known that dim \(P_{\Omega}(M)=1\) or 2 [see \textit{A. Ancona}, J. Math. Pures Appl. IX. Ser. 63, 215-260 (1984; Zbl 0509.31006)]. \textit{M. Benedicks} has studied this question assuming that F is contained in a hyperplane [Ark. Mat. 18, 53-72 (1980; Zbl 0455.31009)]. The present paper deals with the following questions: 1) If n-1 dimensional Lebesgue measure of F is \(>0\), does there exist at least one point M such that dim \(P_{\Omega}(M)=2?\) 2) Is the criterion of Benedicks (loc. cit.) to determine dim \(P_{\Omega}(M)\) valid in the present circumstances? 3) If \(\Omega '\supset \Omega\), does it follow that dim \(P_{\Omega '}(M)\leq \dim P_{\Omega}(M)\), \(\forall M\in \partial \Omega '?\) The author answers 1) affirmatively. As for 2) and 3), the answer is also affirmative for a modification of the Benedicks criterion provided certain geometric conditions of regularity are imposed on \(\Omega\). These conditions are satisfied for hypersurfaces of class \(C^ 2\). Harnack type inequalities at the boundary play an important role. Examples involving the presence of angles are given.
Mathematics Subject Classification ID: 31C35
Mathematics Subject Classification ID: 31B25
zbMATH DE Number: 4161095
zbMATH Keywords: Martin boundary
zbMATH Keywords: cone of positive harmonic functions
zbMATH Keywords: criterion of Benedicks
zbMATH Keywords: Harnack type inequalities
reviewed by: Edwin J. Akutowicz
MaRDI profile type: Publication
cites work: Principle de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien
cites work: Une propriété de la compactification de Martin d'un domaine euclidien
cites work: Régularité d'accès des bouts et frontière de Martin d'un domaine euclidiens
cites work: Positive harmonic functions vanishing on the boundary of certain domains in \(R^ n\).
cites work: On the existence of a largest subharmonic minorant of a given function
cites work: Fatou-Naïm-Doob limit theorems in the axiomatic system of Brelot
cites work: Q5581926
cites work: Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel
cites work: Estimates of harmonic measure
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label: Frontière de Martin d'un domaine de \({\mathbb{R}}^ n\) dont le bord est inclus dans une hypersurface lipschitzienne. (Martin boundary of a dommain in \({\mathbb{R}}^ n\) whose boundary is contained in a Lipschitzian hypersurface)
description: scientific article; zbMATH DE number 4161095