On the boundary behavior of the Dirichlet solutions at an irregular boundary point
Teruo Ikegami · Osaka City University (Osaka City University) · 1984
Introduction.In the classical potential theory, O. Frostman [2] investigated the boundary behavior of the Dirichlet solution Hf for continuous boundary data/ at an irregular boundary point x of a bounded domain U of R".And it was revealed that the cluster set of Hf at x is a segment with a possible exception.In other words, the cluster set of harmonic measures at x has two extreme points -the Dirac measure S x and the balayaged measure βζ u .A generalization of this result was given by Constantinescu-Cornea [1] in an axiomatic setting in a more comprehensive context.Recently, J. Lukes-J.Maly [6] considered this problem in a relatively compact open subset of a harmonic space.The present paper is a contribution to this problem under a resolutive compactification.Let X be a ^-harmonic space with countable base in the sense of Constantinescu-Cornea [1] and X* be a resolutive compactification.Let U be an open set of X.The closure U of U in X* is a resolutive compactification of £7.Suppose that QU=(0\U) Π^X"Φ0.For a sequence \b k } converging to x^QU and satisfying 8ξ u ->εξ u , the harmonic measure of U at b k converges to a measure λ*.If x is irergular for C7, \ x enjoyes remarkable properties stated in Theorem 7, which has a counterpart with the results of Lukes-Mary [6] and Hyvϋnen [3], and is connected with a version of maximal sequences considered by Smyrnόlis [7].In view of the work of Lukes-Maly, we can decide the structure of the cluster set 37^ of harmonic measures and reveal that the type of 3Zf is a local property.We can also conclude the same result for the cluster set of the normalized Dirichlet solutions.