本项目研究由偏微分方程中诱导出的一些集合泛函,如微分算子的第一特征值，capacity等定义的Minkowski问题或Lp-Minkowski问题。这些问题不仅有重要的理论研究价值，而且在三维图像重构和形状设计中也有重要的应用。通过本项目，我们拟解决问题的解的存在性，唯一性和正则性等基本问题。所使用的方法主要是非线性泛函分析中的变分方法，或者偏微分方程中的先验估计的方法。这两种方法在现有的研究文献中表现得非常有效。项目的特色和创新性主要表现在，研究内容的前沿性、新的研究内容的引进以及研究视角的变革等三方面。

In this program,we study Minkowski or Lp-Minkowski problems ,which are defined by domain functional from partial differential equation,such as the principle eigenvalue of differential operator,capacity and so on.These problems are important not only in the Mathematical theory,but also in the applications of reconstruction of three dimensional bodies, or of shape design.The main purpose of this program are going to obtain existence，uniqueness and the regularity result for the problems under consideration.The methods we employ are variational method from nonlinear functional analysis, or the a priori estimate method from nonlinear partial differential equation.These two methods work well in the available references. The innovation of this program include the following three aspects: the topic is in the frontier of the Minkowski theory, new topics are introduced and new research point of view is proposed.