2003 Fiscal Year Final Research Report Summary
Dirichlet space and analysis of harmonic map over the space of Gromov-Hausdorff limit spaces
| Project/Area Number |
13640220
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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 |
Global analysis
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| Research Institution | Yokohama City University |
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Principal Investigator |
KUWAE Kazuhiro Yokohama City University, Graduate school of integrated Science, Associate Professor, 総合理学研究科, 助教授 (80243814)
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| Co-Investigator(Kenkyū-buntansha) |
OTSU Yukio Kyushu University, Graduate School of Mathematics, Associate Professor, 数理学研究院, 助教授 (80233170)
SHIOYA Takashi Tohoku University, Graduate School of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (90235507)
OGURA Yukio Yokohama City University, faculty of Science and Engineering Saga University, Professor, 理工学部, 教授 (00037847)
MACHIGASHIRA Yoshiroh Osaka Kyoiku University, Faculty of Education, Associate Professor, 教育学部, 助教授 (00253584)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Dirichlet form / Harmonic map / Variational convergence / Asymtotic relation / Gamma convergence / Mosco convergence / CAT(0)-space / Gromov-Hausdorff convergence / Alexandrov spaces / Calabi s strong maximum principle |
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| Research Abstract |
(1)The study of variational convergence over metric measured space : This result is a joint work with Professor Takashi Shioya, who is an associate professor of Graduate School of Mathematical Institute, Tohoku University. We introduce a notion called asymptotic relation over a direct sum of metric spaces, which includes the notion of Gromov-Hausdorff convergence as an example. We discuss several notions of functionals over it, for example, Gamma convergence, Mosco convergence and compact convergence and so on. We give a sufficient condition for the compact convergence of functionals. We also prove a sufficient condition for the convergence of resolvents of energy functionals over CAT(0)-spaces.
(2)The study on the stochastic representation of semigroups obtained from a non-symmetric perturbation : This study is a joint work with Professors P.J.Fitzsimmons, Z.Q.Chen and T.S.Zhang. Consider a symmetric regular Dirichlet form and the associated Hunt process admitting jumps of its sample paths. We consider a non-symmetric perturbation by use of locally square integrable martingale additive functionals and a continuous additive functional of finite variation. We prove that the corresponding semigroup has a stochastic representation in terms of time reverse operator on sample paths.
(3)The study of Calabis type strong maximum principle : We give a stochastic proof of an extension of E.Calabi's strong maximum principle in the framework of strong Feller diffusion processes associated with local regular semi-Dirichlet forms. Our results can be applicable to the Gromov-Hausedorff limit space over a family of compact Riemannian manifolds with uniform lower bounds of Ricci curvature and uniform bounds of diameter.
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