1989 Fiscal Year Final Research Report Summary
Research on orders of singularities for solutions to partial differential equations
| Project/Area Number |
63540115
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Kyoto University, Faculty of Science |
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Principal Investigator |
MIYATAKE Sadao Kyoto University, Faculty of Science, Associate Professor, 理学部, 助教授 (10025447)
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| Project Period (FY) |
1988 – 1989
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| Keywords | microlocal order / singularity / distribution / Fourier integral operator / canonical mapping / hyperbolic equation / phase function |
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| Research Abstract |
In this article we define microlocal orders of singularities for distributions and state theorems concerning their important properties. By microlocal orders of singularities we mean not only positions but also directions of singularities. As an application we can clarify how the order of singularity changes by operating Fourier integral operator. For small time t the solution to hyperbolic pseudo- differential equation of order one is described by P_<phi>u , where u is the initial data , phi(x,eta) the phase function and p(x,eta) the amplitude function. Then it holds OS(P_<phi>u;x, xi ) = OS( u; y, eta ) Here OS(u;y, eta ) stands for the order of singularity for u at y in the direction (eta)/(ta1). The mapping (y, eta ) -> (x, xi) is given by the canonical relation generated by phi(x,eta) i.e. y= phi_<eta>(x,eta) and xi= phi_x(x,eta)
Finally we explain OS(u,x_o,xi_o). For f * epsilon' we note OS(f) = inf{r*R, f * H^<-r>} and define OS(f; x_o,xi_o) = inf OS(a(D,X)f) a * S^0, (x_o,xi_o) * (supp a) where (x_o,xi_o) * (supp a) means that a does not vanish in the xi_o - direction. Then it holds the following theorem of Darboux type: For given positive number epsilon there exists delta>O such that OS(f;x_o,xi_o) <less than or equal> OS(s(D,X)f) <less than or equal> OS(f;x_o,xi_o) + epsilon holds for any a * S^0 whose support is contained in delta conic neighborhood of (x_o,xi_o). This inequality plays an important role to prove the first equality.
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Research Products
(2 results)
