Minkowski问题是凸几何分析与积分几何中非常重要的问题，它一直是几何学家所关注的焦点。本项目利用容量(capacity)、Laplacian第一特征值、扭转刚性(torsional rigidity)等与Dirichlet问题相关的变分泛函(variational functionals)研究几何学中的Minkowski问题。主攻以下几个问题：1.深入研究关于以上泛函的Minkowski问题，使其在更多领域中得到应用；2.探索关于以上泛函的对数Minkowski问题；3.研究以上泛函的对偶Brunn-Minkowski不等式，丰富其理论及应用；4.进一步完善已建立的对偶Orlicz-Minkowski问题，寻求更一般测度的对偶Orlicz-Minkowski问题的解。这些结论对发展和完善凸几何分析的Brunn-Minkowski理论都非常重要，在理论和应用上具有重要意义。

The Minkowski problem is a very important problem in convex geometric analysis and integral geometry, and receives great attention from many geometric mathematics. The project aims to study the Minkowski problem by combining three variational functions associated with the Dirichlet problem: capacity, the first eigenvalue of the Laplace operator, and torsional rigidity. The main problems are as follows: 1. We will investigate the Minkowski problem associated with the three functions so that it can be applied in more research areas; 2. We will consider the.Log-Minkowski problem on the three functions; 3. To study the dual Brunn-Minkowski inequalities for the three functions and to enrich their theory and application; 4. We will study the established dual Orlicz-Minkowski problem and seek the solution to dual Orlicz-Minkowski problem for a more general measure. These conclusions are important for the development and improvement of Brunn-Minkowski theory in convex geometric analysis.