| Co-Investigator(Kenkyū-buntansha) |
MIZUTA Yoshihiro Hiroshima Univ. Dept. of Math. Professor, 総合科学部, 教授 (00093815)
SUGIE Jitsuro Shimane Univ. Dept. of Math. Professor, 総合理工学部, 教授 (40196720)
YAMASKI Maretsugu Shimane Univ. Dept. of Math. Professor, 総合理工学部, 教授 (70032935)
HARA Tadayuki Osaka Pref. Univ. Dept. of Math. Professor, 工学部, 教授 (20029565)
MURATA Minoru Tokyo Inst. Tech. Dept. of Math. Professor, 理学部, 教授 (50087079)
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| Research Abstract |
A domain whose boundary lies in a hyperplane is referred to as a Denjoy domain. More generally, we say that a domain is of Denjoy type if it is included in a domain bounded by graphs. The Martin boundary of a Denjoy type domain is more complicated than that of a Denjoy domain. However, if we restrict our attention to the set of positive harmonic functions of finite order, then we can show a result similar to a Denjoy domain.
The number of minimal Martin boundary points over each boundary point of a John domain is finite; moreover, it is estimated in terms of a John constant. If a John constant is sufficiently close to one, then there are at most two minimal Martin boundary points lie over a Euclidean boundary point. Furthermore, a condition for the number of minimal Martin boundary points to be one is given for domains given as unions of convex sets.
John domains, uniform domains and uniformly John domains are characterized provided their boundaries satisfy the capacity density condition. We show that a certain lower estimate of harmonic measures yields the Johnness and vice versa; that the lower estimate as well as a uniform boundary Harnack principle gives a necessary sufficient condition for a domain to be uniform. Moreover, a uniformly John domain is characterized by a uniform boundary Harnack principle with respect to the internal metric.
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