Study on variational problems, optimization problems and nonlinear partial differential equations

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400180
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: KURATA Kazuhiro 首都大学東京, 理工学研究科, 教授 (10186489)
• Co-Investigator(Renkei-kenkyūsha): JIMBO Shuichi 北海道大学, 大学院理学研究院, 教授 (80201565) ; TANAKA Kazunaga 早稲田大学, 大学院理工学術院, 教授 (20188288) ; SHIBATA Tetsutaro 広島大学, 大学院工学研究科, 教授 (90216010) ; SHIBATA Masataka 東京工業大学, 大学院理工学研究科, 助教 (90359688)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 変分問題 / 最適化問題 / 固有値問題 / 非線形楕円型偏微分方程式 / パターン形成

Outline of Final Research Achievements

In this research, we studied the structure of solutions of nonlinear elliptic
partial differential equations arising in mathematical models of quantum phenomena and pattern formation problems of the mathematical biology. Especially, we investigated the precise asymptotic behavior of eigenvalues of the Laplacian with the mixed boundary condition on a thin domain　and the precise asymptotic behavior of the least energy of a nonlinear variational problem associated with the Bose-Einstein condensation. We also studied the structure of global minimizers for 3-component FitzHugh-Nagumo system and the parameter range of existence and non-existence of non-constant steady patterns for several nonlinear rection-diffusion systems, e.g., the prey-predator model with a cross-diffusion effect.

Free Research Field

偏微分方程式