Une propriété de la compactification de Martin d'un domaine euclidien
The author studies properties of the minimal point in the Martin boundary of a Euclidean domain. Let \(\Omega\) be a connected open subset of Euclidean space \({\mathbb{R}}^ n\) (n\(\geq 2)\) having a Green function G and denote by \({\hat \Omega}\) the Martin compactification of \(\Omega\) relative to the sheaf of ordinary harmonic functions. Let \(\Delta ={\hat \Omega}-\Omega\) and \(\Delta_ 1\) be the set of minimal points of \(\Delta\). The main result is the following: If \(\Omega\) contains an open ball B(x,r) \((r>0)\) and \(y\in \partial B(x,r)\cap \partial \Omega\), then the point \(P_ t=y+t(x-y)\) converges to a point \(\zeta_ 0\) of \(\Delta_ 1\) in \({\hat \Omega}\) as t tends to 0 in [0,1). In general if a filter \({\mathcal F}\) on B converges nontangentially to a point of \(\partial \Omega \cap \partial B\), then \({\mathcal F}\) converges to a minimal point in the Martin boundary \(\Delta\) of \(\Omega\). The key to showing such results is the Harnack inequality. Theorem 1 is also essential: Let \(B_{\alpha}\) be the intersection of B(x,r) and B(y,\(\alpha\) r) for \(x\in \Omega\), \(y\in \partial B(x,r)\) and \(\alpha\in (0,1/2]\), and denote by \(\Sigma_{\alpha}\) the segment \([y+\alpha (x- y),y+2(x-y)]\). Then there exists a constant \(C>0\), independent of n, such that, for all \(\alpha\in (0,1/20]\), for all \(P,P'\in B_{\alpha}\) and \(Q\in \Sigma_{10\alpha}\), \[ G(P,Q)/G(P,P_{2\alpha})\leq CG(P',Q)/G(P',P_{2\alpha}). \] In the latter half of the paper, some applications of this result are treated. Finally the author discusses a conjecture of R. S. Martin - that in general \(\Delta ={\bar \Delta}_ 1\) in \({\hat \Omega}\)- and obtains the negative solution. An earlier paper [ibid. 28, 169-213 (1978; Zbl 0377.31001)] by the athor is quoted as a key reference for this paper.
scientific article; zbMATH DE number 3946570