viscosity solutions of nonlinear partial differential equations with singularities

具有奇点的非线性偏微分方程的粘度解

• 批准号: 10640119
• 负责人: ISHII Katsuyuki
• 金额: $ 1.86万
• 依托单位: Kobe University of Mercantile Marine
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640119/
• 关键词: subdifferential; nonlinear PDE; vviscosity solutions; motion by mean curvature; radially symmetric solutions; 退化放物型偏微分方程式; 退化楕円型偏微分方程式

In this project, I considered the existence, uniqueness and stability of viscosity solutions of nonlinear partial differential equations (PDE 's in short) with singularities and their applications of some approximate problems. I had some results on the motion of planar polygons with singular curvature and its application to an approximation for the planar motion of a simple closed curve by its curvature. I also showed that a version of an algorithm, which was proposed by Bence, Merryman and Osher in 1992, can be applied to approximate the motion by mean curvature with right-angle boundary condition in a bounded domain.I studied elliptic/parabolic PDE's with nonlinear terms of the spatial gradient. I classifed completely the interaction between the growth properties of nonliner terms and the uniqueness classes for viscosity solutions and proved the existence of viscosity solutions in such classes. I also treated nonlinear second order ellitpic PDE's with subdifferential. Using the definition of the subdifferential, we modified the notion of the usual viscosity solutions and obtained the uniqueness, existence and stability.Maruo mainly studied the radially symmetry of continuous viscosity solutions of Dirichlet problem for nonlinear degenerate elliptic PDE's. He gave the necessary and sufficient condition which assures that the continuous viscosity solutions are radially symmetric. It seems that this condition is optimal. He also obtained the existence and uniqueness of bounded radial viscosity solutions and those of unbounded ones in the whole space.

本课题研究了具有奇异点的非线性偏微分方程（PDE）黏性解的存在性、唯一性和稳定性及其在一些近似问题中的应用。我在具有奇异曲率的平面多边形的运动方面得到了一些结果，并将其应用于简单闭合曲线的平面运动的曲率近似。我还展示了Bence， Merryman和Osher在1992年提出的算法的一个版本，可以应用于有界域中具有直角边界条件的平均曲率近似运动。我研究了具有非线性空间梯度项的椭圆/抛物型偏微分方程。完整地分类了非线性项的生长性质与黏性解的惟一类之间的相互作用，并证明了该类中黏性解的存在性。本文还用次微分法处理了非线性二阶椭圆偏微分方程。利用次微分的定义，修正了通常的粘性解的概念，得到了其唯一性、存在性和稳定性。Maruo主要研究了非线性退化椭圆型偏微分方程的连续粘滞解的径向对称性。他给出了连续粘度解径向对称的充分必要条件。这个条件似乎是最优的。他还得到了有界径向粘度解和无界径向粘度解在整个空间中的存在唯一性。

项目成果

Hitoshi Ishii: "An Approximation scheme for motion by mean curvature with right-angle boundary condition"SIAM J.Math.Anal.. 33. 369-389 (2001)

Hitoshi Ishii：“直角边界条件下平均曲率运动的近似方案”SIAM J.Math.Anal.. 33. 369-389 (2001)

Kenji Maruo: "Radial viscosity solutions of the Dirichlet problems for semilinear elliptic equations"Osaka J.Math.. 38. 737-757 (2001)

Kenji Maruo：“半线性椭圆方程狄利克雷问题的径向粘度解”Osaka J.Math.. 38. 737-757 (2001)

Katsuyuki Ishii: "Unbounded viscosity solutions of nonlinear second order PDE's"Adv. Math. Sci. Appl.. 10. 689-710 (2000)

Katsuyuki Ishii：“非线性二阶偏微分方程的无界粘度解”Adv。

Katsuyuki Ishii: "Nonlinear second order elliptic PDE's with subdifferential"Adv.Math.Sci.Appl.. 12(in press). (2002)

Katsuyuki Ishii：“具有次微分的非线性二阶椭圆偏微分方程”Adv.Math.Sci.Appl.. 12（印刷中）。

Hitoshi Ishii and Katsuyuki Ishii: "An approximation scheme for motion by mean curvature with right angle boundary condition"SIAM J.Math.Anal.. (to appear).

Hitoshi Ishii 和 Katsuyuki Ishii：“直角边界条件下平均曲率运动的近似方案”SIAM J.Math.Anal..（即将出版）。