Research on viscosity solutions of differential equations and their applications

微分方程粘性解及其应用研究

• 批准号: 12440044
• 负责人: ISHII Hitoshi
• 金额: $ 9.92万
• 依托单位: Waseda University (2001-2002)Tokyo Metropolitan University (2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440044/
• 关键词: Viscosity solutions; Optimal control; Curvature flow; Hamilton-Jacobi equations; Gauss curvature flow; Stochastic optimal control; Level set approach; Uniformly elliptic equations; 凸化ガウス曲率流; 離散確率近似; 緩和; 積分微分方程式; 特異摂動; 適正粘性解; 半連続粘性解; 退化楕円形方程

The results obtained in our project are : (1) In 1974, W. Firey proposed the Gauss curvature flow as a mathematical model of the wearing process of a stone rolling on the beach by wave. This model assumes that the stone has a convex shape. In our research we considered the case when a stone is not convex. We introduced the convexified Gauss curvature flow which models the wearing process of a nonconvex stone rolling on the beach and established the level set approach based on viscosity solutions method. (2) We introduced weak solutions to the integral equation which describes the convexified Gauss curvature flow, proved that the uniqueness of the weak solution for the Cauchy problem, and proved its existence by a discrete stochastic approximation. (3) We studied a general stochastic optimal control problem with state constraint and proved, under relatively weak assumptions, the Lipschitz continuity and Holder continuity of the associated value function, that the value function satisfy the corresponding Bellman equation in the viscosity sense and that the state constraint problem for the Bellman equation has a unique viscosity solution. (4) We introduced the notion of proper viscosity solutions of a wide class of first-order partial differential equations including the Burgers equation, proved the unique existence of proper viscosity solutions for the class of equations, which may not have divergence form, and established the convergence of the approximation by the vanishing viscosity method to proper viscosity solutions in the sense of convergence of graphs with respect to the Hausdorff distance.

我们的研究结果是：（1）1974年，W. Firey提出了高斯曲率流作为海滩上石头在波浪作用下滚动磨损过程的数学模型。该模型假设石头具有凸形形状。在我们的研究中，我们考虑了石头不是凸的情况。引入了凸化高斯曲率流模型来模拟非凸石头在海滩上滚动的磨损过程，建立了基于粘性解的水平集方法。(2)在描述凸化Gauss曲率流的积分方程中引入弱解，证明了Cauchy问题弱解的唯一性，并通过离散随机逼近证明了其存在性. (3)研究了一般的带状态约束的随机最优控制问题，在较弱的假设条件下，证明了相应值函数的Lipschitz连续性和保持器连续性，并证明了该值函数满足相应的Bellman方程在粘性意义下，Bellman方程的状态约束问题存在唯一的粘性解. (4)引入了包括Burgers方程在内的一类一阶偏微分方程的真粘性解的概念，证明了这类方程的真粘性解的唯一存在性，在图关于Hausdorff的收敛性意义下，建立了用消失粘性方法逼近真粘性解的收敛性距离

项目成果

Y.Giga: "Viscosity solutions with shocks"Comm. Pure Appl. Math.. 55・4. 431-480 (2002)

Y.Giga：“带有冲击的粘度解决方案”Comm. 55・480（2002）

石井仁司: "A Dirichlet type problem for nonlinear degenerate elliptic equations arising in time-optimal stochastic control"Adv.Math.Sci.Appl.. 10・1. 329-352 (2000)

Hitoshi Ishii：“时间最优随机控制中出现的非线性简并椭圆方程的狄利克雷型问题”Adv.Math.Sci.Appl.. 10・1 (2000)。

Yoshikazu Giga: "Viscosity solutions with shocks"Comm.Pure Appl.Math.. 55. 431-480 (2002)

Yoshikazu Giga：“具有冲击的粘度解决方案”Comm.Pure Appl.Math.. 55. 431-480 (2002)

Hitoshi Ishii and Kazufumi Shimano: "Asymptotic analysis for a class of infinite systems of first-order PDE : nonlinear parabolic PDE in the singular limit"Comm. Partial Differential Equations. 28-1/2. (2003)

Hitoshi Ishii 和 Kazufumi Shimano：“一阶 PDE 无限系统的渐近分析：奇异极限下的非线性抛物线 PDE”Comm。

Shigeaki Koike and Toshimi Takahashi: "Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients"Adv. Differential Equations. 7-4. 493-512 (2002)

Shigeaki Koike 和 Toshimi Takahashi：“关于具有可测量成分的完全非线性均匀椭圆偏微分方程的粘度解的正则性的评论”Adv.