On the study of the theory of viscosity solutions and its new developments

论粘度解理论的研究及其新进展

• 批准号: 16340032
• 负责人: KOIKE Shigeaki
• 金额: $ 10.5万
• 依托单位: Saitama University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16340032/
• 关键词: viscosity solution; fully nonlinear equation; degenerate elliptic equation; uniformly elliptic equation; calculus of variation; Harnack inequality; Mathematical Finance; optimal control; ハルナックの不等式; 変分問題

The Aleksandrov-Bakelman-Pucci maximum principle for Lp-viscosity solutions of fully nonlinear second order uniformly elliptic/parabolic partial differential equations with possibly superllinear growth terms of the first derivatives, unbounded coefficients, unbounded inhomogeneous terms has been established under appropriate hypotheses in two research papers with A. Swiech. Some counter-examples have been presented when there are no hypotheses.Perron's method has been first applied to Lp-viscosity solutions of fully nonlinear elliptic partial differential equations by introducing semicontinuous Lp-visosity solutions.For several nonlinear variational inequalities arising in Mathematical Finance, optimal controls have been constructed by showing that associated value functions admit enough regularity in research papers with H. Morimoto, and H. Morimoto and S. Sakaguchi.

在适当的假设下，建立了具有一阶导数可能超线性增长项、无界系数、无界非齐次项的全非线性二阶一致椭圆/抛物型偏微分方程的线性-黏性解的Aleksandrov-Bakelman-Pucci极大原理。在没有假设的情况下，提出了一些反例。通过引入半连续的lp -黏度解，首次将Perron方法应用于全非线性椭圆型偏微分方程的lp -黏度解。对于数学金融中出现的几个非线性变分不等式，在H. Morimoto和H. Morimoto和S. Sakaguchi的研究论文中，通过表明相关值函数具有足够的规律性，构造了最优控制。

项目成果

Optimal Rate of Convergence of the Bence-Merriman-Osher Algorithm for Motion by Mean Curvature

• DOI: 10.1137/04061862x
• 发表时间: 2005
• 期刊: SIAM J. Math. Anal.
• 作者: K. Ishii

Maximum principle and existence of Lp-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

• DOI: 10.1007/s00030-004-2001-9
• 发表时间: 2004-12
• 期刊: Nonlinear Differential Equations and Applications NoDEA
• 作者: Shigeaki Koike;Andrzej Świch

Maximum principle for fully nonlinear PDEs via the iterated comparison function method

通过迭代比较函数法实现完全非线性偏微分方程的极大值原理

• 发表时间: 2006
• 作者: K.;Oguiso;S. Koike;S. Koike
• 通讯作者: S. Koike

数学辞典・第4版「粘性解」

数学词典，第 4 版“粘性溶液”

• 发表时间: 2007
• 作者: 石井仁司;小池茂昭
• 通讯作者: 小池茂昭

Maximum principle for $L^p$-viscosity solutions of fully nonlinear equations with unbounded coefficients

具有无界系数的完全非线性方程的 $L^p$-粘度解的极大值原理

• 发表时间: 2007
• 作者: Kohji;Matsumoto;Takashi;Nakamura;Hirofumi;Tsumura;Hideo NAGAI;中村憲 他4名;R.Hosoya and H.Suzuki;Shigeaki Koike
• 通讯作者: Shigeaki Koike