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.

在这个项目中，我考虑了具有奇性的非线性偏微分方程粘性解的存在唯一性和稳定性及其在一些近似问题中的应用。我得到了一些关于具有奇异曲率的平面多边形运动的结果，并将其应用于简单闭合曲线的平面运动的曲率逼近。我还证明了Bence，Merryman和Osher在1992年提出的一个算法的一个版本，该算法可以用有界域上具有直角边界条件的平均曲率来逼近运动。对非线性项的增长性与粘性解的唯一性类之间的相互作用进行了完整的分类，并证明了粘性解在这类类中的存在性。我还用次微分方法处理了二阶非线性偏微分方程组。利用次微分的定义，修正了通常粘性解的概念，得到了解的唯一性、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组Dirichlet问题连续粘性解的径向对称性，给出了保证连续粘性解是径向对称的充要条件。看起来这个条件是最理想的。他还得到了全空间有界径向粘性解和无界粘性解的存在唯一性。

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）。