A study on nonlinear degenerate elliptic partial differential equations

非线性简并椭圆偏微分方程的研究

• 批准号: 02640150
• 负责人: ISHII Hitoshi
• 金额: $ 0.51万
• 依托单位: CHUO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1990
• 资助国家: 日本
• 起止时间: 1990 至 1991
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-02640150/
• 关键词: Viscosity Solutions; Degenerate Elliptic Partial Differential Equations; Optimal Control; Hamilton-Jacobi Equations; Mean Curvature Flow; 非線形偏微分方程式; ハミルトン・ヤコビ・ベルマン方程式; 退化楕円型方程式; 微分ゲ-ム; 退化楕円形方程式

We have studied the existence and uniqueness of viscosity solutions of nonlinear elliptic partial differential equations and its applications. The results are the following: We have given a formulation of the maximum principle for semicontinuous functions and proved it. The result asserts the existence of a generalized second order derivatives for semicontinuous functions at their maximum points when they have a special form. The result makes the theory of viscosity solutions quite complete. We have studied generalized flows of surfaces by mean curvature and showed that for the proof of the uniqueness of the generalized flows, it is important to use the functions of 4th power as a test function in the standard argument of uniqueness. We also have shown that if the initial surface is convex then so is the surface in the future. We have studied monotone systems of partial differential equations. We have studied second order elliptic partial differential equations in Hilbert spaces and established the existence and uniqueness theorem of viscosity solutions.

研究了非线性椭圆偏微分方程粘性解的存在性、唯一性及其应用。结果如下： 我们给出了半连续函数极大值原理的表述并证明了它。结果断言当半连续函数具有特殊形式时，在极大点处存在广义二阶导数。这一结果使得粘度解的理论相当完整。我们通过平均曲率研究了曲面的广义流，并表明，为了证明广义流的唯一性，在唯一性标准论证中使用四次方函数作为检验函数非常重要。我们还表明，如果初始表面是凸面，那么未来的表面也是凸面。我们研究了偏微分方程的单调系统。研究了希尔伯特空间中的二阶椭圆偏微分方程，建立了粘性解的存在唯一性定理。

项目成果

石井 仁司,小池 茂昭: "Viscosity solutions for monotone systems of secondーorder elliptic syetems" Communications in partical differential egnations. 16. 1095-1128 (1991)

Hitoshi Ishii，Shigeaki Koike：“二阶椭圆系统的单调系统的粘度解决方案” 部分微分方程中的通信 16. 1095-1128 (1991)。

H.Ishii, S.Koike: "Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games" Funk. Ekvac.34. 143-155 (1991)

H.Ishii、S.Koike：“切换游戏中出现的非线性二阶椭圆偏微分方程系统的粘度解”Funk。

H.Ishii: "Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces" J. Func. Anal.105. 301-341 (1992)

H.Ishii：“希尔伯特空间中一类 Hamilton-Jacobi 方程的粘度解”J. Func。

石井 仁司: "Fully nonlinear obligue deriuative prob lems for nonlinear secondーorder elliptic PDE's" Duke Mathematical Journal. 62. 633-661 (1991)

Hitoshi Ishii：“非线性二阶椭圆 PDE 的完全非线性斜导数问题”杜克数学杂志 62. 633-661 (1991)。

M.G.Crandll 石井 仁司: "The maximum principle for semicontinuous functions" Differential and integral eguations. 3. 1001-1014 (1990)

M.G.Crandll Hitoshi Ishii：“半连续函数的极大值原理”微分和积分阐述 3. 1001-1014 (1990)。