关于特殊拉格朗日方程的外狄利克雷问题的研究

The exterior Dirichlet problem for fully nonlinear elliptic equation is an important problem in the field of fully nonlinear equations. A series of important developments have been made in this area in recent years. The types of equations involved include the Monge-Amp\`{e}re equation and the Hessian Equation and so on...Based on our recent successful research on the Hessian quotient equation, the purpose of our research in this project is to solve the problem of the existence and uniqueness of the solution of the exterior Dirichlet problem of a special Lagrangian equation...The special Lagrangian equation originates from the calibrated geometry, the theory of the minimal surface equation, the Calabi-Yau space theory and the string theory in theoretical physics. From the formal point of view, it is a fully nonlinear elliptic equation more nonlinear and more complex than the Monge-Amp\`{e}re equation, the Hessian equation and the Hessian quotient equation. It is a basic problem to investigate the existence and uniqueness of the solution of the exterior Dirichlet problem of a special Lagrangian equation, and this research is also important in understanding the asymptotic behavior of the solution of the equation at infinity. We will develop a systematic method to deal with the related nonlinear problems, which not only plays very important roles in the study of the nonlinearity of the special Lagrangian equation, but also enriches the theories and techniques of general partial differential equations. Furthermore, the study of this project is of great value in the understanding of the geometric problems and physical phenomena controlled by this equation.

完全非线性椭圆方程的外狄利克雷问题是完全非线性方程领域的重要问题，这方面近多年已取得一系列重要研究进展，具体方程类型包括蒙日-安培方程和海森方程等。..在我们近期关于海森商方程成功研究的基础上，本项目我们旨在解决特殊拉格朗日方程的外狄利克雷问题的解的存在唯一性问题。..特殊拉格朗日方程起源于校准几何，同极小曲面方程理论，卡拉比-丘空间理论以及理论物理学中的弦理论都有着比较深入的联系。同时它也是比蒙日-安培方程，海森方程和海森商方程更复杂和更非线性的方程。研究方程外狄利克雷问题解的存在唯一性，是一个基本问题；同时它也有助于我们理解该方程的解在无穷远处的渐近性态。在研究过程中我们将发展出一套处理相关非线性问题的系统的方法，这对于拓展我们对特殊拉格朗日方程非线性型的认知，对于丰富一般偏微分方程理论和技术，进而对于理解由特殊拉格朗日方程所描述的几何和物理现象都具有非常重要的价值。

完全非线性方程的外狄利克雷问题是完全非线性方程领域的重要问题, 这方面近多年已取得一系列重要研究进展, 具体方程类型包括蒙日-安培方程和海森方程等...在我们前期关于海森商方程成功研究的基础上, 本项目执行期间, 我们解决了特殊拉格朗日方程的外狄利克雷问题的解的存在唯一性问题, 和其他方面一些结果...特殊拉格朗日方程起源于校准几何, 同极小曲面方程理论, 卡拉比-丘空间理论以及理论物理学中的弦理论都有着比较深入的联系. 同时它是比蒙日-安培方程, 海森方程和海森商方程更复杂和更非线性的方程. 研究方程外狄利克雷问题解的存在唯一性, 是一个基本问题; 同时它也有助于我们理解该方程的解在无穷远处的渐近性态. 在研究过程中我们发展出了一套处理相关非线性问题的系统的方法, 这对于拓展我们对特殊拉格朗日方程非线性类型的认知, 对于丰富一般偏微分方程理论和技术, 进而对于理解由特殊拉格朗日方程所描述的几何和物理现象都具有非常重要的价值.