2011 Fiscal Year Final Research Report
Study on the structure of nonnegative solutions for parabolic equations and the perturbation theory of elliptic operators
| Project/Area Number |
21540164
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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 | Tokyo Institute of Technology |
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Principal Investigator |
MURATA Minoru 東京工業大学, 大学院・理工学研究科, 教授 (50087079)
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| Co-Investigator(Kenkyū-buntansha) |
SHIGA Hiroshige 東京工業大学, 大学院・理工学研究科, 教授 (10154189)
UCHIYAMA Kouhei 東京工業大学, 大学院・理工学研究科, 教授 (00117566)
MIYAMOTO Yasuhito 東京工業大学, 大学院・理工学研究科, 助教 (90374743)
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| Project Period (FY) |
2009 – 2011
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| Keywords | 解析学 / 関数方程式 / 関数解析学 / 確率論 |
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| Research Abstract |
We investigated the structure of nonnegative solutions to parabolic equations in cylinders on Riemannian manifolds, and gave explicit integral representation formulas for any solutions under the general and optimal condition that the constant function 1 is a semismall perturbation of the associated elliptic operator ; whose geometric characterization was also given in the case of the heat equation on rotationally symmetric Riemannian manifolds. Furthermore, by using the characterization and giving a sharp sufficient condition for the uniqueness of nonnegative solutions to the Cauchy problem, we determined the structure of nonnegative solutions to the heat equation on rotationally symmetric Riemannian manifolds.
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