Geometry and Analysis of non-linear Partial Differentail Equations

非线性偏微分方程的几何与分析

• 批准号: 08454011
• 负责人: IZUMIYA Shyuichi
• 金额: $ 5.44万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454011/
• 关键词: Hamilton-Jacobi equations; Conservation laws; Viscosity sokutions; Shock waves; Fascet surfaces; Ginzburg-Landau equations; Lagrangian stability; Cohomology; ギンツブルクランダウ方程式; シンプレクティック多様体; 非線形偏微分方程式; ハミルトン・ヤコビ方程式; ルジャンドル組み紐; シュレディンガー方程式

In this research project, we established the calssification of the singularities of solution surfaces of quasi-homogeneous first order partial differential equations, viscosity solutions of Hamilton-Jacobi equations with one-space variable and multivalued solutions of conservation laws which are a part of the main purpose. Moreover, we extend the theory ofviscosity solutions to the case when the second order non-degenerate equation with non-local effects (one-space variable). This research is important to describe the crystal growth with fascet surfaces. We also give some iimportant new examples of Riemannian manifolds with integrable geodesic flows.On the other hand, we have shown the existence of stable solutions for Ginzburg-Landau equation in a rotain domain. We have given a characterization of the symplectic and Lagarangian stablity of isotropic submanifolds with corank one by using a kind of transversality theorem. As a result on Algebraic and Geometric Topology we have developed an elementary tools for calculating the cohomology of heyper eliptic mapping class groups over finite fields.These results are contained in the areas of the border of Geometry and Analysis. We expect to apply these results for studying Partial Differentail Equations in near future.

在该研究项目中，我们确定了准同质一阶部分偏微分方程的溶液表面的奇异性，汉密尔顿 - 雅各比方程的粘度解决方案具有单空间变量和多价保护法的粘度解决方案，这是主要目的的一部分。此外，当二阶非脱位方程具有非本地效应（一个空间变量）时，我们将愿景解决方案的理论扩展到了案例。这项研究对于描述筋膜表面的晶体生长很重要。我们还提供了一些具有整合地测量流的Riemannian歧管的新示例。另一方面，我们已经在Rotain域中展示了Ginzburg-Landau方程的稳定溶液。我们通过使用一种横向定理给出了各向同性亚曼属的符号和拉加朗加稳定性的表征。结果，我们开发了一种基本工具，用于计算有限字段上的Heyper Eliptic映射类组的共同点。这些结果包含在几何和分析边界的区域中。我们希望在不久的将来应用这些结果来研究部分不同的方程。

项目成果

T, FUKUI: "Motion of graph bymonsmooth weighted curvature" Proc.of the first World congress of monlinear andysis.92. I. 47-56 (1996)

T，FUKUI：“Monsmooth 加权曲率图的运动”Proc.of 第一届世界单线性安迪斯大会。92。

N.KAWAZUMI: "An infinitesimal approach to the stable cohomokgy of the mootuli of Riemann snrfaces" Topology and Teich muller Spaces,World Scientific. 1. 79-100 (1996)

N.KAWAZUMI：“黎曼面的 mootuli 的稳定同调的无穷小方法”拓扑和 Teich muller 空间，世界科学。

S.JIMBO: "Stable solutions with zero to the Ginzburg-Landau equation with Neumann boundary condition" J.Differential Equations. 128. 596-613 (1996)

S.JIMBO：“具有诺依曼边界条件的 Ginzburg-Landau 方程的零稳定解”J.Differential Equations。

G.ISHIKAWA: "Trans versalrties for Lagronge singularities of isotropic mappings of co rankone." Banach Center Publications. 33. 93-104 (1996)

G.ISHIKAWA：“corankone 各向同性映射的拉格朗日奇点的横向关系。”

G, ISHIKAWA: "Acviterion for fimte topological determinacy of smooth map-germs" Proc.London Math.Soc.74・3. 662-700 (1997)

G，石川：“光滑地图细菌的有限拓扑确定性的Acviterion”Proc.London Math.Soc.74・3（1997）。