2001 Fiscal Year Final Research Report Summary
Potential theory in a domain with fractal boundary
| Project/Area Number |
11640154
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Ochanomizu University |
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Principal Investigator |
WATANABE Hisako Ochanomizu Univ., Graduate School of Humanities and Sciences, Professor, 大学院・人間文化研究科, 教授 (70017193)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Michie Ochanomizu University, Faculty of Sciences, Professor, 理学部, 教授 (30017206)
MATSUZAKI Katsuhiko Ochanomizu University, Faculty of Sciences, Assistant Professor, 理学部, 助教授 (80222298)
TAKEO Fukiko Ochanomizu University, Faculty of Sciences, Professor, 理学部, 教授 (40109228)
YOSHIDA Hidenobu Chiba Univ. Graduate School of Natural Sciences, Professor, 大学院・自然科学研究科, 教授 (60009280)
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| Project Period (FY) |
1999 – 2001
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| Keywords | fractal boundary / double layer potentials / Whitney decomposition / Besov spaces / Besov norms / maximal functions / uniform domains / boundedness of operators |
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| Research Abstract |
We consider the boundary-value problems in a domain D with fractal boundary. It often occurs that an operator K on the Besov space on the boundary is bounded with respect to the Besov norms. We can prove the boundedness of an operator from δD to δD in the following method.
(1) We extend a function defined on δD to R^n by using an extension operator E.
(2) The Besov norm of f is estimated by (∫_D |▽f(x)|^<Pλ>dx)^<1/P>, where δ(x) is the distance from x to δD.
(3) Instead of the boundedness of K we prove the boundedness of an operator F from D to the outside of D with respect to suitable norms by using the maximal functions between D and the outside of D.
We proved the boundedness of an operator K, which is important to solve the Dirichlet problem by using double layer potentials.
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