拉格朗日平均曲率方程是指欧式空间或其他度量空间中预定一个函数梯度图平均曲率的二阶完全非线性椭圆方程, 是极小(平均曲率为零)拉格朗日方程的一个推广. 近年来, 与之相关的伯恩斯坦定理, 解在无穷远点的渐近性, 解在孤立奇点的性质已经或正在成为国际数学界的热点. 本课题组将在申请人和项目组成员已有工作的基础上, 充分运用具体方程的几何解释, 综合使用偏微分方程, 微分几何, 调和分析, 复分析和非线性分析及代数等多种数学手段, 从特殊到一般地讨论这些方程解在奇点（含无穷远点）处的性质. 预计首先对欧式空间中的拉格朗日平均曲率方程方面取得进展, 完成经典伯恩斯坦定理在外区域的各种推广, 获得解在无穷远点的渐近性. 另一方面, 将研究拉格朗日平均曲率方程在孤立奇点附近的性质. 与此平行, 开始对其他度量下的拉格朗日平均曲率方程进行多角度的讨论.

The Lagrangian mean curvature equation is a second order fully nonlinear elliptic equation, which prescribes the mean curvature of gradient graph in a Euclid or other metric space. It is a generalization of special Lagrangian equation (the case of zero mean curvature). In recent years, the related Bernstein theorem, the asymptotic property of solutions at infinity, the behavior of isolated singularities has been or is becoming a hot issue in the international mathematics research. Based on the work done in this field, the group will discuss the properties at singularities or infinity of the solutions of these equations from the special to the general, by making full use of the geometrical background of the equations, and various mathematical methods such as partial differential equation, differential geometry, harmonic analysis, complex analysis, nonlinear analysis and algebra. It is expected to make some progress on the Lagrangian mean curvature equation firstly in Euclid space, and then to finish various generalizations of the classical Bernstein theorem in the exterior domain to obtain the asymptotic behavior of the solution at infinity. On the other hand, they will study the properties of Lagrangian mean curvature equations near isolated singularities. Parallel to this, they begin a multi-angle discussion of Lagrangian mean curvature equations in other metric space.