增广Hessain方程斜微商问题的正则性

This project aims to study a kind of augmented Hessian partial differential equations. We mainly study the their C^2 and C^{2,α} regularity for their associate boundary value problems such as linear oblique problem, second boundary value problem, capillary boundary problem, etc. We shall discuss how the ellipticity of the equation, the regularity of the matrix, the nonlinearity of boundary condition and the convexity of the domain affect the a priori estimates and the classical existence. We shall establish the necessary and sufficient conditions of the classical solvability of the linear oblique problem, and obtain the global Schauder estimate for nonlinear oblique problem. Furthermore, we study the related geometric problem, such as the nonlinear Yamabe problem with boundary.. Both the augmented Hessian equations and their oblique boundary value conditions arise in applications such as doptimal transportation, optics and geometry problems. Therefore, it has important significance in studying the oblique derivative problem for this kind of equations. Comparing with the Dirichlet problem, the gradient estimate and the second derivative estimate on the boundary of the oblique problem are more difficult. We shall conquer these difficulties and study the delicate global regularity of oblique derivative problems.. By developing a series of new methods, further research in this direction can enrich the theory of fully nonlinear elliptic equations, which provides the fundamentals in solving the geometric problems and real-world problems.

本项目讨论一类增广Hessian方程，研究其线性斜微商问题、第二类边值问题等的全局C^2和C^{2,α}正则性。我们将讨论椭圆性的强弱，矩阵性质的优劣，边值非线性程度的高低，区域凸性的强弱，对解的先验估计和正则性的本质影响，建立线性斜微商问题经典解存在的充分必要条件，以及非线性斜微商问题的全局Schauder估计，进而研究相关的几何问题，例如共形几何中带边的非线性Yamabe问题等。. 增广Hessian方程和斜边值条件均来源于最优运输、光学和几何问题，对于这类问题的研究具有重要意义。相较于Dirichlet边值问题，斜微商问题在边界上的一阶导数、二阶导数的先验估计更困难。我们将克服这些困难对斜微商问题的全局正则性进行深入细致的研究。. 开展此项目的研究，发展一系列新的研究方法和手段，可丰富完全非线性椭圆偏微分方程斜边值问题的理论，并为解决几何和实际问题提供理论基础。

本项目主要讨论一类增广Hessian方程斜边值问题解的存在性、唯一性和正则性，该研究可应用于最优运输、几何光学和共形几何等问题。主要研究成果如下：研究了一类非退化增广Hessian型方程斜边值问题，得到了经典解的存在唯一性；研究了一类退化增广Hessian型方程斜边值问题，证明了粘性解的存在性、唯一性和正则性；证明了退化Monge-Ampere方程Neumann边值问题强解的存在性和唯一性；建立了Monge-Ampere方程改进形式的Pogorelov估计；讨论了一般增广Hessian方程的Dirichlet边值问题经典解的存在唯一性；研究了共形k-Hessian及Hessian商不等式全局正解的不存在性；建立了无穷Laplace型方程的李普希兹估计；建立了一些抛物、超抛物方程的Harnack不等式。