Research on the theory of viscosity solutions and its applications

粘度解理论及其应用研究

• 批准号: 15340051
• 负责人: ISHII Hitoshi
• 金额: $ 7.74万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340051/
• 关键词: viscosity solutions; curvature flow; mathematical finance; optimal control; level set approach; asymptotic problems; 凸化ガウス曲率流

We proposed and proved the effectiveness of singular diffusions in the vertical direction in the level set approach to first-order partial differential equations (pde for short). We established the strong maximum principle to viscosity solutions of fully nonlinar elliptic pde including the minimal surface eqaution. We builded an example of fully nonlinear uniformly elliptic pde for which the maximum principle does not holds, and established the maximum principle, Holder regularity, and the solvability of the Dirichlet problem for such nonlinear pde under suitable hypotheses. We introduced the convexified Gauss curvature flow, formulated the level set approach to its generalizations, and established existence and uniqueness of solutions of the pde which appears in the level set approach. We also introduced a stochastic approaximation scheme to the generalized convexified Gauss flow and proved its convergence. We proved on a mathematical basis the occurrence of Berg's effect when the crystal shape is a cylinder. For the BMO (Bence-Merrima-Osher) scheme, we gave a new proof of its convergence to the mean curvature flow and the optimal estimate on the rate of convergence. We proved the convergence the asymptotic solutions as time goes to infinity of solutions of parabolic pde with the Ornstein-Uhlenbeck operator. We analized the simultaneous effects of homogenization and vanishing viscosity in periodic homogenization of uniformly elliptic pde. We proved existence and uniqueness of the limit in the zero-noise of certain h-path processes and established existence and uniqueness of the Monge-Kantorovich problem with a quadratic cost. Regarding mathematical finance, we studied optimal stopping time problems and risk-sensitive portfolio optimization problems for general factor models and constructed their optimal strategies. We analized the asymptotic behavior of solutions of p-Laplace equations as p goes to infinity in a fairly general setting.

提出并证明了水平集方法求解一阶偏微分方程（简称偏微分方程）中垂直方向奇异扩散的有效性。建立了包含极小曲面方程的完全非线性椭圆型偏微分方程粘性解的强极大值原理。建立了一个完全非线性一致椭圆型偏微分方程极大值原理不成立的例子，在适当的假设下，建立了这类偏微分方程的极大值原理、保持器正则性和Dirichlet问题的可解性.我们引入了凸化高斯曲率流，将水平集方法推广到它的推广，并建立了水平集方法中出现的偏微分方程解的存在唯一性.我们还对广义凸化高斯流引入了一种随机逼近格式，并证明了它的收敛性。本文从数学上证明了当晶体形状为圆柱形时，贝格效应的发生。对于BMO（Bence-Merrima-Osher）格式，给出了其收敛于平均曲率流的一个新的证明和收敛速度的最优估计.证明了带有Ornstein-Uhlenbeck算子的抛物型偏微分方程解的渐近解随时间趋于无穷远的收敛性。分析了均匀椭圆偏微分方程周期均匀化中均匀化和粘性消失的同时作用。证明了一类h-路过程在零噪声下极限的存在唯一性，建立了二次费用Monge-Kantorovich问题的存在唯一性.在金融数学方面，研究了一般因子模型的最优停时问题和风险敏感投资组合优化问题，并构造了它们的最优策略。在相当一般的情形下，我们分析了p-Laplace方程解在p趋于无穷大时的渐近性态。

项目成果

On breakdown of solutions of a constrained gradient system of total variation.

关于全变分约束梯度系统解的分解。

• 发表时间: 2004
• 期刊: Bol.Soc.Parana.Mat. (3) 22, no.1
• 作者: Y.Giga;H.Kuroda
• 通讯作者: H.Kuroda

The principle of symmetric criticality for non-differentiable mappings.

不可微映射的对称临界性原理。

• 发表时间: 2004
• 期刊: J.Funct.Anal. 214, no.2
• 作者: J.Kobayashi;M.Otani
• 通讯作者: M.Otani

Anisotropic curvature flow in a very thin domain.

非常薄的域中的各向异性曲率流。

• 发表时间: 2003
• 期刊: Indiana Univ.Math.J. 52, no.2
• 作者: M.Arisawa;Y.Giga
• 通讯作者: Y.Giga

Topological degree for (S)+-mappings with maximal monotone perturbations and its applications to variational inequalities

• DOI: 10.1016/j.na.2004.07.007
• 发表时间: 2004-10
• 期刊: Nonlinear Analysis-theory Methods & Applications
• 影响因子: 1.4
• 作者: Jun Kobayashi;M. Otani

Degree for subdifferential operators in Hilbert spaces

希尔伯特空间中次微分算子的次数

• 发表时间: 2004
• 期刊: Advances in Mathematical. Sciences and Applications 14巻1号
• 作者: 小林純;大谷光春
• 通讯作者: 大谷光春