对偶Minkowski问题及若干非线性偏微分方程的研究

Minkowski problem is an important problem that is related to geometric measures in convex geometric analysis. In general, the solution of a Minkowski problem amounts to solving a degenerate fully nonlinear partial differential equation. Nonlinear elliptic partial differential equations, parabolic partial differential equations, nonlinear equations in differential geometry, and partial differential equations in materials and applied mathematics are the core research topics of modern mathematics. The research scope of this project will involve nonlinear elliptic partial differential equations related to differential geometry, especially Monge ampere equation related to equidistant embedding and optimal mass transportation, geometry and Analysis on Heisenberg Group, and nonlinear partial differential equations related to dual Minkowski problem.

Minkowski问题是凸几何分析中与几何测度相关的著名问题。一般来说，Minkowski问题的解决在某种意义上对应于非退化完全非线性的偏微分方程的解。非线性椭圆偏微分方程、抛物偏微分方程、微分几何中的非线性方程、以及材料与应用数学中的偏微分方程是现代数学的核心研究主题。本项目研究范围将涉及微分几何相关的非线性椭圆偏微分方程，特别是与等距嵌入和最优质量运输问题有关的Monge-Ampere方程，Heisenberg群上的几何与分析，及与对偶Minkowski问题相关的非线性偏微分方程等内容。

项目主要是研究非线性偏微分方程的边值问题以及给出几何应用，对于完全非线性椭圆$\Sigma_k(u)=0$的外问题，麻希南首先与张德凯做了一个存在性,关键是一致梯度估计与一致二阶导数估计，郭路军老师在科大合作访问期间把我们的梯度估计推广到Sigma_(k,p)(u)=0方程，那是有几何意义的完全非线性退化奇异方程。现在二阶估计还没有做出，我2023年计划利用今年余下的国家基金再邀请他在来科大访问。