Studies on the applications of the theory of viscosity solutions to some singular perturbation problems

粘性解理论在某些奇异摄动问题中的应用研究

• 批准号: 14540117
• 负责人: ISHII Katsuyuki
• 金额: $ 1.66万
• 依托单位: Kobe University (2004)神戸商船大学 (2002-2003)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540117/
• 关键词: motion by mean curvatute; numerical algorithm; viscosity solutions; degenerate elliptic partial differential equations; 退化楕円型方程式; 退化放物型方程式

Katsuyuki Ishii studied a numerical algorithm for motion by mean curvature, which is proposed by Bence, Merriman and Osher and obtained the following results1.I gave a proof of convergence showing how the mean curvature flow equation is derived from this algorithm. (This is a joint work with Yoko Goto and Takayoshi Ogawa.)2.I obtained the rate of convergence of this algorithm in the case of smoth motion by mean curvature. I also showed the optimality in the case of a circle evolving by curvature.Kenji Maruo studied semilinear degenerate elliptic partial differential equations in the plane. He obtained the following3.Assuming that the coefficients of the equation are radially symmetric, he proved that, under some growth conditions at infinity, the continuous viscosity solutions are radially symmetricYasuo Kageyama obtained the rate of convergence and some properties of modified Bernstein polynomials

Katsuyuki Ishii研究了Bence， Merriman和Osher提出的平均曲率运动的数值算法，得到了以下结果1。我给出了收敛性的证明，展示了如何从这个算法推导出平均曲率流方程。（这是与后藤洋子和小川隆芳的合作作品。）我通过平均曲率得到了该算法在光滑运动情况下的收敛速度。我还展示了圆按曲率演化的最优性。Kenji Maruo研究了平面上的半线性退化椭圆型偏微分方程。他得到了以下内容：他假设方程的系数是径向对称的，证明了在无穷远的某些增长条件下，连续粘度解是径向对称的

项目成果

A note on zeros of Lagrange interpolation polynomial of the function 1/(z-c)

关于函数 1/(z-c) 的拉格朗日插值多项式零点的注解

• 发表时间: 2003
• 期刊: Transaction of Japanese Society for Industrial and Applied Mathematics Vol.13, No.3
• 作者: Kenji Maruo;Yoshihito Tomita;Yasuo Kageymma
• 通讯作者: Yasuo Kageymma

Nonlinear second order elliptic PDEs with subdifferential

具有次微分的非线性二阶椭圆偏微分方程

• 发表时间: 2002
• 期刊: Advances in Mathematical Sciences and Applications Vol.12 No.1
• 作者: Kenji Maruo;Yoshihito Tomita;Yasuo Kageymma;Katsuyuki Ishii
• 通讯作者: Katsuyuki Ishii

K.Maruo, Y.Tomita: "Unbounded radially symmetric viscosity solutions of semilinear degenerate elliptic equations"Math. Japonicae. 8. 107-123 (2003)

K.Maruo，Y.Tomita：“半线性简并椭圆方程的无界径向对称粘度解”数学。

K.Maruo, Y.Tomita: "Remarks on Viscosity solutions of the Divichlet problem for Quasilinear degenerate elliptic equations"Funkcial. Ekvac.. 45. 89-101 (2002)

K.Maruo、Y.Tomita：“关于拟线性简并椭圆方程 Divichlet 问题的粘度解的评论”Funkcial。

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