完全非线性椭圆方程解的边界正则性

In this project, we will study the dependence of the regularity of solutions of fully nonlinear elliptic equations on the boundary conditions of the equations and the geometric properties of the boundary of the domains. Our study will go on simultaneously two aspects, that is, we will study not only what is the optimal regularity of solutions under prescribed boundary conditions of the equations and boundary geometric conditions, but also what boundary conditions of the equations and boundary geometric conditions are needed in order to obtain the desired regularity of solutions. We mainly focus on the following three fundamental problems: (1) The boundary differentiability and the boundary C^{k,α} regularity of solutions of fully nonlinear elliptic equations; (2) The Wiener criterions of fully nonlinear elliptic equations, that is the solutions satisfy the boundary conditions pointwisely; (3) The integrability of solutions and their derivatives of fully nonlinear elliptic equations near the boundary. The targets of this project contains two parts: one is to solve the above three problems and to present a system of related theorems; the other is to develop new techniques and methods for studying boundary regularity. These theorems, techniques and methods will be widely applied in geometric design in engineering, optimal control, inverse problems and free boundary problems. Previously achievements of ours will lay a solid foundation for this project.

本项目将从正反两个方面研究完全非线性椭圆方程解的边界正则性对方程边界条件和区域边界几何性质的依赖性，即不仅要研究在给定边界条件和区域边界几何性质的条件下解的最佳边界正则性，而且还将探索获得解的特定边界正则性所需区域边界的几何性质和方程的边界条件。项目将着重研究以下三个方面的问题:（1）完全非线性椭圆方程解的边界可微性与C^{k,α}正则性；（2）完全非线性椭圆方程解在边界逐点满足边界条件的Wiener判别准则；（3）完全非线性椭圆方程解及其导数在边界附近的可积性。项目的目标不仅要解决上述3个问题，系统地给出相关的理论，而且更进一步还将发展出一套新的研究边界正则性的方法。项目所得到的理论和发展的方法将在工程中的几何设计、最优控制、反问题、自由边界问题等方面有着广泛的应用。我们在前期已取得的成果为本项目的完成奠定了坚实的基础。

本项目研究了完全非线性椭圆方程解的边界正则性，圆满地完成了项目申请时所制定的研究计划，从正反两个方面给出解的边界正则性对方程边界条件和区域边界几何性质的依赖性。主要结论有四个方面：1、证明了完全非线性椭圆方程在斜边界条件下的Harnack不等式，Jensen唯一性定理，以及C^α，C^(1,α)和C^(2,α)估计。2、给出了外区域上完全非线性椭圆方程以及锥上非散度椭圆方程解在无穷远处和原点处的渐进行为。3、得到边界具有奇异性或退化性的k-Hessian方程解在边界的渐进行为。4、提出了椭圆方程部分方向齐次化问题，并证明了解在齐次化方向上和非齐次化方向上正则性的不同。通过对本项目的研究，在ARMA、Adv.Math.、JMPA、JFA、CV、JDE等国际著名杂志上共发表学术论文22篇。在本项目的支持下，招收1名博士后、5名博士生和4名硕士生。依托本项目的支持，项目组成员共参加国内、国际会议26个，参加人数为50余人次；出境参加国际会议6人次。项目于2019年11月举行了“西安2019偏微分方程最新进展研讨会”，有72位来自全国各地的专家代表出席了此次研讨会。项目各项经费使用符合项目申请时所制定的经费预算。