2007 Fiscal Year Final Research Report Summary
Variational study of nonlinear problems
| Project/Area Number |
17540205
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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 | Waseda University |
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Principal Investigator |
TANAKA Kazunaga Waseda University, Faculty of Science and Engineering, Professor (20188288)
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| Co-Investigator(Kenkyū-buntansha) |
OTANI Mitsuharu Waseda University, Faculty of Science and Engineering, Professor (30119656)
YAMAZAKI Masao Waseda University, Faculty of Science and Engineering, Professor (20174659)
YAMADA Yoshio Waseda University, Faculty of Science and Engineering, Professor (20111825)
SHIBATA Tetsutaro Hiroshima University, Graduate School of Engineering, Professor (90216010)
KURATA Kazuhiro Tokyo Metropolitan University, Graduate School of Science and Engineering, Professor (10186489)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Variationa Problems / Nonlinear Differential Equations / Singular Perturbation / Hamiltonian systems / Minimax methods |
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| Research Abstract |
We study nonlinear elliptic partial differential equations and Hamiltonian systems via variational meth-ods. We put emphasis on singular perturbation problems.
1. We study the existence of high frequency solutions-families of solutions whose numbers of spikes or layers increase to ∞ as the singular perturbation parameter ε goes to 0. We give the existence and the characterization of such families for 1 dimensional elliptic problems including nonlinear Schrodinger equations, Allen-Cahn equations, Fisher equations and Girerer-Meinhardt systems. Especially for Girerer-Meinhardt systems, we introduce and analyze a limit equation using adiabatic invariants. We also give a precise estimate of the number of positive solutions of nonlinear Schrodinger equations.
2. We also study a singular perturbation problem for -ε^2△μ+V(χ)μ =g(μ) in R^N. Under very general conditions on g(μ), which is related to the work of Berestycki, Gallouet-Kavian, we prove the existence of a concentrating solution for N=1,2.
3. We also study the prescribed energy problem for singular first order Hamiltonian systems. We suc-ceed to obtain the existence of periodic orbit under conditions which generalize the strong force condition of Gordon. We remark that our condition is given as a property of the energy surface S={(q, p);H(q, p) =E} not on the Hamiltonian H(q, p).
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