非局部方程边值问题解的凸性理论和带边k-Yamabe问题

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。

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.