| Project/Area Number |
12440040
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Shimane University |
|---|
Principal Investigator |
AKIKAWA Hiroaki Shimane Univ. Dept. of Math. Professor, 総合理工学部, 教授 (20137889)
|
|---|
| 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)
|
|---|
| Project Period (FY) |
2000 – 2002
|
| Project Status |
Completed (Fiscal Year 2002)
|
| Budget Amount *help |
¥6,600,000 (Direct Cost: ¥6,600,000)
Fiscal Year 2002: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2001: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2000: ¥2,900,000 (Direct Cost: ¥2,900,000)
|
|---|
| Keywords | Denjoy domain / Martin boundary / harmonic function / John domain / convex set / uniform domain / boundary Harnack principle / capacity density condition / Dirichlet解 / 多調和関数 / 放物型方程式 / 時間遅れをもつ方程式 / 境界値 / 漸近挙動 / 整関数 / ポテンシャル / 調和 / ソボレフ関数 / 境界条件 / 放物型 / 安定性 / 関数微分方程式 / ボルテラ型微分方程式 |
|---|
| 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.
|
