具有约束项的Hessian方程解的先验估计

In this project, we study the a priori estimates of the solutions of Hessian equations. Inspired by the methods of Trudinger, Wang and Chou, our research mainly contain these aspects as follows: (1)Weakening the conditions of the constraint f, namely, under the case that the index of constraint f is strictly larger than 1, we give the method of a priori estimates of the solutions of elliptic Hessian equations; (2) Constructing new auxiliary functions, we establish the accurate a priori estimates of some parabolic Hessian equations, and deduce the corresponding existence theory; (3)We study the applications of Hessian Sobolev inequality and Moser-Trudinger inequality in the a priori estimates of a Hessian equation, with elliptic type and parabolic type. A priori estimates of the solutions is the main problem of fully nonlinear partial differential equations. Under more general cases, the establishment of a priori estimates theory of Hessian equation will promote the study of many problems such as the existence of solutions, regularity theory and variational theory of Hessian equation.

本项目研究Hessian方程解的先验估计问题. 受Trudinger、Wang、Chou等人方法的启发, 本项目主要研究: (1)减弱约束项f的条件, 即在约束条件指数严格大于1的情形下给出椭圆型Hessian方程解的先验估计方法; (2) 构造新的辅助函数, 给出更一般情形下的抛物型Hessian方程解的精确的先验估计, 并建立相应的解的存在性理论; (3)研究Hessian-Sobolev不等式和Moser-Trudinger不等式在椭圆型和抛物型k-Hessian方程的先验估计理论中的应用. 解的先验估计是完全非线性方程理论研究的主要问题, 建立更加一般情形下Hessian方程解的先验估计理论, 将推动Hessian方程理论比如解的存在性、正则性以及变分理论等重要问题的研究.

根据本项目的计划书，我们主要做了以下方面的工作。第一，通过学习Xu, Wang的方法，我们减弱约束项f的条件，在更弱的条件下讨论椭圆型Hessian方程解的先验估计问题；第二，通过构造新的辅助函数，我们给出了一类含有积分项的抛物型Hessian方程解的先验估计，并最终得出其解的存在性，这类方程的解的存在性在Hessian-Sobolev不等式的证明中都有应用，所以我们事实上使之前该不等式的证明显得完整。同时，我们试图结合上述两方面的工作，探讨Hessian-Sobolev不等式和Moser-Trudinger不等式在Hessian方程的解的先验估计问题中的应用。