Fundamental theory for viscosity solutions of fully nonlinear equations and its applications

2016 Fiscal Year Final Research Report

• Project/Area Number: 23340028
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tohoku University
• Principal Investigator: Koike Shigeaki 東北大学, 理学研究科, 教授 (90205295)
• Co-Investigator(Kenkyū-buntansha): 小川 卓克 東北大学, 理学(系)研究科(研究院), 教授 (20224107) ; 石井 克幸 神戸大学, 海事科学研究科(研究院), 教授 (40232227) ; 石井 仁司 早稲田大学, 教育・総合科学学術院, 教授 (70102887) ; 長井 英生 関西大学, 工学部, 教授 (70110848) ; 三上 敏夫 津田塾大学, 学芸学部, 教授 (70229657) ; 石毛 和弘 東北大学, 理学(系)研究科(研究院), 教授 (90272020) ; 岡部 真也 東北大学, 理学(系)研究科(研究院), 准教授 (70435973)
• Project Period (FY): 2011-04-01 – 2016-03-31
• Keywords: 粘性解 / 最大値原理 / 比較原理 / ハルナック不等式 / 完全非線形 / 障害問題 / 非局所作用素

Outline of Final Research Achievements

We obtained comparison principle for unbounded viscosity solutions of degenerate elliptic PDE with superlinear gradient terms. We presented a representation formula for viscosity solutions of integro-differential equations of Isaacs type. We established the local maximum principle fro Lp-viscosity solutions of fully nonlinear uniformly elliptic PDE with unbounded coefficients to the first derivatives. We discussed regularity and large time behavior of viscosity solutions of integro-differential equations with coercive first derivative terms. We obtained existence and uniqueness of entire solutions of fully nonlinear elliptic equations with superlinear growth in the first derivatives. We showed the Lipschitz continuity of viscosity solutions of mean curvature flow equations with bilateral obstacles.

Free Research Field

非線形偏微分方程式