2006 Fiscal Year Final Research Report Summary
Study of Green function of higher order / fractional order differential equations from a viewpoint of a reproducing kernel theory
| Project/Area Number |
17540175
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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 University of Technology |
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Principal Investigator |
TAKEMURA Kazuo Tokyo University of Technology, Research Assistant, メディア学部, 助手 (60367216)
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| Co-Investigator(Kenkyū-buntansha) |
KAMETAKA Yoshinori Osaka University, Emeritus Professor, 大学院基礎工学研究科, 名誉教授 (00047218)
NAGAI Atsushi Nihon University, Lecturer, 生産工学部, 講師 (90304039)
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| Project Period (FY) |
2005 – 2006
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| Keywords | Green function / Reproducing kernel / Sobolev inequality / Best constant / Ordinary differential equation |
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| Research Abstract |
(2005) We constructed Green functions under various boundary conditions and showed that the Green functions are reproducing kernels of suitable Hilbert spaces. Based on this fact, we succeeded in calculation of the best constant and the best function for Sobolev inequality by examining a diagonal value of Green function in a detailed manner.
We calculated the best constant of a Sobolev inequality corresponding to several boundary value problems including Diriclet type, Neumann type and the periodic type conditions for a string bending problem. If the corresponding eigenvalue problem has a nonpositive eigenvalue, we constitute a generalied Green function by the so-called symmetric orthogonalization method by imposing the solvability and orthogonality condition to the boundary value problem.
(2006) We calculated concretely the best constant of a Sobolev inequality corresponding to boundary value problems for 2M-th order differential operator, which contains clumped type, Diriclet type, Neumann type, a free end, a periodic type condition. In particular, the best constant of Dirichlet, Neumann and periodic boundary condition is found and expressed by means of Bernoulli polynomials and Riemann zeta function. This result give a variational meaning of Riemann zeta function. In the other 2 cases, we calculated the best constant of a Sobolev inequality by examining a diagonal value of Green function.
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Research Products
(8 results)
