Geometry and Analysis of non-linear Partial Differentail Equations

在本研究项目中，我们建立了拟齐次一阶偏微分方程解面奇异性的分类、单空间变量Hamilton-Jacobi方程的黏性解和守恒律的多值解，这是主要目的的一部分。此外，我们将粘性解的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）的情况。该研究对描述具有束状表面的晶体生长具有重要意义。我们还给出了具有可积测地线流的黎曼流形的一些重要的新例子。另一方面，我们证明了金兹堡-朗道方程在固定域上稳定解的存在性。利用一种横截定理，给出了具有corank 1的各向同性子流形的辛稳定性和Lagarangian稳定性的刻画。在代数和几何拓扑的基础上，我们开发了一个计算有限域上超椭圆映射类群上同调的基本工具。这些结果包含在几何和分析的边界区域中。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.