2012 Fiscal Year Final Research Report
Mathematical inverse problems based on analysis of complex geometrical optics solutions and the applications to science and engineering
| Project/Area Number |
21740107
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
Basic analysis
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| Research Institution | Doshisha University |
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Principal Investigator |
TAKUWA Hideki 同志社大学, 理工学部, 准教授 (80403111)
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| Project Period (FY) |
2009 – 2012
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| Keywords | 逆問題 / 複素幾何光学解 / 擬リーマン幾何学 / カーレマン評価式 |
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| Research Abstract |
We have studied the special solutions to the mathematical inverse problems. These solutions are called complex geometrical solutions, in short, CGO solutions. It was known that we could succeed to apply CGO solutions with linear complex phase functions to many problems. Recently new CGO solutions with nonlinear complex phase functions have been studied. But this new approach was restricted to the problems about elliptic equations as Laplace equations. So no one has understood the meaning of CGO solutions with nonlinear phase functions in general cases. In this research program we have studied new CGO solutions with nonlinear phase functions which can be applicable to general equations including hyperbolic equations. More precisely, we can derive new nonlocal Carleman estimates. By using this estimate we can study Lorentian metric and operators associated it. This is the new inverse problem related to hyperbolic equations.
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[Remarks] 多久和英樹, 常微分方程式の境界値問題の固有関数展開, 数学セミナー, vol. 51 no12, 614, 2012年12月, pp20-pp23
