Deepening of the theory of viscosity solutions and its applications

2015 Fiscal Year Final Research Report

• Project/Area Number: 23244015
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Waseda University
• Principal Investigator: ISHII Hitoshi 早稲田大学, 教育・総合科学学術院, 教授 (70102887)
• Co-Investigator(Kenkyū-buntansha): OTANI Mitsuharu 早稲田大学, 理工学術院, 教授 (30119656) ; NAGAI Hideo 関西大学, 工学部, 教授 (70110848) ; GIGA Yoshikazu 東京大学, 数理科学研究科, 教授 (70144110) ; KOIKE Shigeaki 東北大学, 理学研究科, 教授 (90205295) ; MIKAMI Toshio 津田塾大学, 学芸学部, 教授 (70229657) ; MITAKE Hiroyoshi 広島大学, サステナブル・ディベロップメント実践研究センター, テニュアトラック講師 (90631979)
• Co-Investigator(Renkei-kenkyūsha): YAMADA Naoki 福岡大学, 理学部, 教授 (50030789) ; ISHII Katsuyuki 神戸大学, 海事科学研究科, 教授 (40232227) ; ICHIHARA Naoyuki 青山学院大学, 理工学部, 准教授 (70452563) ; FUJITA Yasuhiro 富山大学, 理工学研究部, 教授 (10209067)
• Project Period (FY): 2011-04-01 – 2016-03-31
• Keywords: 関数方程式 / 粘性解 / 最適制御 / 曲面の発展方程式 / 完全非線形偏微分方程式 / 弱ＫＡＭ理論

Outline of Final Research Achievements

We investigated the asymptotic problems of partial differential equations such as the long-time asymptotic behavior of solutions of Hamilton-Jacobi equations and viscous Hamilton-Jacobi equation, the vanishing discount problem, and obtained many important new pieces of knowledge regarding these asymptotic problems as well as the theory of viscosity solutions. We developed the basic theory of the existence and uniqueness of solutions for singular diffusion equations and for integral-differential equations. Based on the analysis of solutions of Hamilton-Jacobi-Bellman equations, we established certain estimates on the large-time asymptotic behavior of the minimizing large deviation probabilities, the verification theorem for optimal consumption-investment in a non-complete market model, a new approach to the stochastic optimal transportation problem.

Free Research Field

数学