偏微分方程解的凸性及其几何应用

The study on convexity of the solutions to partial differential equation is a classic subject. It is important to solving geomrtic and analysis problems, also to getting the uniquness, regularity and existence for solutions of partial differential equation. In this project, we will use constant rank theorem to cope with several convexity problems related to elliptic and parabolic equations. We will focus on the convexity of level sets of solution to the Dirichlet boundary value problem with prescribed constant mean curvatures,the geometric properites of travelling wave solutions to Allen-Cahn equations in whole space. We will also explore the correspoding problems on Riemannian manifold which were put forward by Yau, so as to understand the corresponding variants on Riemann manifolds for some important geometric inequalities in Euclidean space. At last, we give an application for convexity: to prove the symmetry results for some kinds of elliptic equations with overdetermined boundary value on convex domains. Finding geometric properties of level sets of solution with degenerate gradient is partly an open problem. In this project we will face this degenarate case.Now we know that constant rank theorem is a kind of microscopic method which can be used to deal with unbounded domian or with Riemannian manifold, however we should explore it as in our project. At last we are looking for a Brunn-Minkowski functional to deal with overdetermined problems.

偏微分方程解的凸性研究是一个经典主题，它在许多几何和分析问题、方程解的唯一性、正则性和存在性等问题中都具有重要意义。常秩定理是处理凸性问题的一个强有力的工具,本项目我们将主要运用该定理来研究以下几个问题：1）常平均曲率方程Dirichlet边值问题解的水平集凸性2）Allen-Cahn 方程在全空间上行波解的几何性态3）丘成桐提出的黎曼流形上的凸性问题，希望推导出欧氏空间著名的几何不等式在黎曼流形上的表达形式4)凸性在几何上的运用：凸区域上几类椭圆方程超定边值问题解的对称性。 问题1)涉及到了梯度退化，该情形下研究方程解的水平集的几何性质是一个公开问题，也是目前已有结果都回避的。另外常秩定理是一种微观方法,可以处理无界区域甚至是黎曼流形上的凸性问题,这是我们将要探索的方向如问题2)和3)。对于问题4)我们关键需要利用凸性来构造关于区域的Brunn-Minkowski泛函。

一、 利用凸性理论证明某些泛函的Brunn-Minkowski不等式，从而研究凸区域上p-laplace方程的超定问题解的对称性；.二、 构造不同辅助函数，在L-p Minkowski问题，Minkowski 空间中 Fuchsian convex surfaces, 预定Hessian曲率方程, 共形k-Hessian方程上分别得到解的Liouville结果；.三、 研究一类最优输运Monge–Ampère方程Dirichlet边值问题，预定平均曲率方程斜边值问题，对偶 Brunn–Minkowski 问题解的先验估计以及存在性。