2022 Fiscal Year Final Research Report
Variational study of non-local nonlinear elliptic equations
| Project/Area Number |
17H02855
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical analysis
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| Research Institution | Waseda University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
小澤 徹 早稲田大学, 理工学術院, 教授 (70204196)
黒田 隆徳 早稲田大学, 理工学術院, 講師(任期付) (00907058)
大谷 光春 早稲田大学, 理工学術院, 名誉教授 (30119656)
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Keywords | 変分問題 / 非局所問題 / 非線形楕円形方程式 / 特異摂動問題 / ミニマックス法 |
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| Outline of Final Research Achievements |
Nonlinear problems with non-local terms appear in many important problems in mathematical physics. Via variational methods we study such nonlinear problems with non-local terms. Especially we show for nonlinear Choquard equations and nonlinear Schroedinger equations with fractional Laplacian the existence and multiplicity of standing waves via minimax approaches. We also study L2-constraint problem and singular perturbation problems for non-local problems. To show our existence results, we develop a new deformation theories, which work under weak Palais-Smale type conditions.
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| Free Research Field |
解析学, 変分問題
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| Academic Significance and Societal Importance of the Research Achievements |
数理物理学に現れる重要な問題は変分構造をもつ. また量子力学に関連する重要な問題には非局所項を伴うものが多い. ここではこれらの問題の数学的構造に注目し, 定常状態に対応する定在波の存在, 多重性の理論的研究を行った. その際, 無限次元空間での変形理論が重要となる. ここでは Palais-Smale 型の弱いコンパクト性の条件のもとで働く変形理論 (deformation theory) を新たに開発し用いている.
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