| Project/Area Number |
13640166
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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 | Nagoya University |
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Principal Investigator |
SUZUKI Noriaki Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (50154563)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIGE Kazuhiro Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (90272020)
MIYAKE Masatake Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70019496)
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| Project Period (FY) |
2001 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2001: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | harmonic function / heat equation / Dirichlet problem / mean value theorem / heat ball |
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| Research Abstract |
We study the continuation and uniqueness for solutions of partial differential equations, by using potential theory. We have the following results.
1. In 2001, we showed a characterization of heat balls by mean value property for temperatures in Proc.Amer.Math.Soc. Then a generalization of it was obtained and published in Suriken Kokyuroku. Based on these results, we start to study the existence of mean value density for temperatures. In particular, a relation with the Dirichlet regularity and the existence of a bounded density or a density with positive infimum are discussed. The development of them is our new object of study.
2. We discueesd an extension of harmonic function on a domain. In the 2 dimensional case our problem is completely solved, but in higher dimensional case there are some problems.
3. We study a polynomial solution of the Dirichlet problem on a domain for the heat equation. In case that a domain is determined by a polynomial with degree less than 3, we obtain a necessary and sufficient condition under which the above problem is solvable. This result is publised in Bull.Aichi Inst.Tech. In case the degree is more than 4 we find that this problem is closely related with an assertion concerning to the zero points of Hermite polynomials.
4. In the connection of uniqueness of a solution of α parabolic operators on a half space, we study the Huygens property and the duality of parabolic Bergman spaces. Our spaces contain the usual harmonic and heat Bergman spaces.
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