Geometry and Analysis of non-linear Partial Differentail Equations

在本研究计划中，我们建立了拟齐次一阶偏微分方程解曲面的奇异性分类，单空间变量Hamilton-Jacobi方程的粘性解和守恒律的多值解，这是主要目的的一部分。此外，我们将粘性解理论推广到具有非局部效应的二阶非退化方程（单空间变量）的情形。该研究对描述具有faset表面的晶体生长具有重要意义。另外，我们还证明了旋转区域上Ginzburg-Landau方程稳定解的存在性。利用一种横截性定理给出了余秩为1的迷向子流形的辛稳定性和Lagarang稳定性的一个刻划。作为代数学和几何拓扑学的一个结果，我们发展了一个计算有限域上Heyper椭圆映射类群的上同调的初等工具，这些结果包含在几何学和分析学的边界部分。我们期望在不久的将来将这些结果应用于研究偏微分方程。

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.