2004 Fiscal Year Final Research Report Summary
Study of nonlinear differential equations via variational methods
| Project/Area Number |
14540216
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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, 理工学術院, 教授 (20188288)
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| Co-Investigator(Kenkyū-buntansha) |
OTANI Mitsuharu Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (30119656)
YAMAZAKI Masao Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (20174659)
KURATA Kazuhiro Tokyo Metropolitan University, Faculty of Urban Environmental Sciences, Professor, 都市教養学部, 教授 (10186489)
SHIBATA Tetsurato Hiroshima University, Graduate School of Engineering, Professor, 大学院・工学研究科, 教授 (90216010)
ISHIWATA Michinori Waseda University, Faculty of Science and Engineering, Research Associate, 理工学術院, 助手 (30350458)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Variational Problems / Nonlinear Differential Equations / Singular Perturbation / Hamiltonian systems |
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| Research Abstract |
We study the existence and multiplicity of solutions of nonlinear differential equations via variational methods. In particular, we study singular perturbation problems.
1.We study the existence and multiplicity of solutions of nonlinear scalar field equations : -Δu+V(x)u=f(u) in R^N. Usually in such a problem global conditions on nonlinearity f(u)(ex.global Ambrosetti-Rabinowitz condition) are required to ensure the existence of solutions. In this study we tried to obtain an existence result without such global assumptions and we find that it is possible if we require sufficiently fast decay of the potential V(x).
2.We also study singular perturbation problem : -Δu+λ^2a(x)u=|u|^<p-1>u in R^N, where a(x)【greater than or equal】0. As a limit problem as λ→∞, a Dirichlet boundary value problem -Δu=|u|^<p-1>u, u|_<∂Ω>=0 in Ω≡{x ∈R^N;a(x)=0} appears. We assume Ω consists of several bounded connected components Ω_1,【triple bond】, Ω_κ and for given solutions u_i(x) of the Dirichlet problem in Ω_i, we try to find a solution u_λ(x) in R^N whose limit is u_i(x) in Ω_i (connecting problem). We succeed to find a solution joining Mountain Pass solutions without non-degeneracy conditions. Also we show that there are infinitely many sign-changing solutions that are connectable with Mountain Pass solutions.
3.For 1-dimensional Allen-Cahn equations and Schrodinger equaitons, we study the characterization of a family of solutions in the setting of singular perturbation. More precisely, we consider a family of solutios with increasing number of layers or spikes. We give a characterization of such a family using "limit enery function" or "envelop function". Conversely for addmissible patterns we construct corresponding families of solutions via variational methods.
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