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)
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.
scientific article; zbMATH DE number 4161095
scientific article