非局部方程边值问题解的凸性理论和带边k-Yamabe问题

Theory of PDEs plays important roles in modern analysis and geometry, and attracts much attention for a long time. The main aims of this project are two folds, i.e. convexity theory of nonlocal PDEs and k-Yamabe problem on bounded domain in Euclidean space. Convexity theory of PDEs is mainly study the fine properties of their solutions. In contrast to the extensively study of convexity theory of local PDEs, there are few results on nonlocal equations up to now. This project will establish the general theory of convexity of solutions and level sets of nonlocal equations. The classical Yamabe problem with boundary is that given a compact manifold with boundary, one asks whether there exists a metric with constant scalar curvature and minimal boundary. The k-Yamabe problem is its natural extension, and it is equivalent to solve a fully nonlinear Hessian type equation with Neumann boundary condition. In the second part of the project, we will study the k-Yamabe problem on bounded domain in Euclidean space by the method of a priori estimates which developed recently for Neumann problem of Hessian equations. Our final aim is to solve real geometric problems.

偏微分方程理论在现代分析学和几何学中扮演着重要角色，对它们的研究一直以来都备受关注。本项目的研究内容包括两部分，即非局部偏微分方程的凸性理论和欧氏空间中有界区域上的高阶Yamabe问题。偏微分方程的凸性理论以解的精细性质为主要研究内容。截止目前，局部方程的凸性理论已有不少结果，但非局部方程的凸性研究刚刚起步。通过本项目的研究，我们希望建立起非局部方程解和水平集凸性的一般理论，以便为非局部版本的几何问题提供凸性方面的理论依据。经典的带边Yamabe问题是说给定一个紧致带边流形，问是否存在共形于边界极小的常数量曲率度量。带边高阶Yamabe问题是其自然推广，本质上这等价于求解带Neumann边界条件的完全非线性Hessian型方程。我们希望将新发展起来的建立Hessian方程Neumann问题先验估计的方法应用到欧氏空间中有界区域上的高阶Yamabe问题中去,解决实在的几何问题。

本项目的研究内容主要包括两部分: 一是非线性椭圆方程解和水平集的凸性理论; 二是几何非线性椭圆方程Neumann问题的可解性. 总体上, 项目进展顺利, 完成期刊论文10篇(含已接收论文), 达到了预期目标. 具体来说, 我们主要得到了下面几方面的结果: (1)欧氏空间中调和函数和极小图水平集曲率估计的新改进; (2)欧氏空间中Torsion函数和Laplace第一特征函数解的幂凸性估计的新改进; (3)椭圆和抛物方程的常秩定理及其在凸性理论上的应用; (4)k-曲率方程的过定问题; (5)拟线性方程和完全非线性方程Neumann问题的可解性.