2004 Fiscal Year Final Research Report Summary
Potential theoretic approach to parabolic equations
| Project/Area Number |
13640186
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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 | Osaka City University |
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Principal Investigator |
NISHIO Masaharu Osaka City University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90228156)
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| Co-Investigator(Kenkyū-buntansha) |
KOMATU Takashi Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80047365)
SAKAN Ken-ichi Osaka City University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70110856)
MASAOKA Hiroaki Kyoto Sangyo University, School of Science, Professor, 大学院・理学研究科, 教授 (30219315)
SUZUKI Noriaki Nagoya University, Graduate School of Mathematics, Associate Professor, 理学部, 教授 (50154563)
SHIMOMURA Katsunori Ibaraki University, School of Science, Associate Professor, 理学部, 助教授 (00201559)
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| Project Period (FY) |
2001 – 2004
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| Keywords | heat equation / mean value property / harmonic function / quasiconformal mapping / Martin boundary / caloric morphism / polytemperature / parabolic equation of fractional order |
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| Research Abstract |
We have studied on parabolic equations and related subjects to obtain the following results :
1. Mean value property.
Suzuki characterized the heat ball by using mean value properties. He had a result on density functions of mean value properties.
2. Quasiconformal mappings.
(1) Sakan improved the Heinz's results for harmonic and quasiconformal mappings
(2) Masaoka showed that the rigidity property of the minimal Martin boundary of Rieman surfaces of Heinz type for quasiconformal mappings.
3. Martin boundary.
Masaoka gave the relation of various classes of harmonic functions with Martin boundary of Rieman surfaces of Heinz type.
4. Caloric morphism
(1) Nishio and Shimomura gave a differential equation which characterizes caloric morphisms between manifolds.
(2) Shimomura classifed caloric morphisms on a certain semi-riemannian manifold.
5. Polytemperature.
Shimomura showed that the Appell transformation is the only map which presearves polytemperatures.
6. Parabolic equation of fractional order
Nishio, Shimomura and Suzuki introduced the notion of parabolic Bergman spaces and obtained the fundamental results on completeness, reproducing kernels, dual spaces, etc.
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