Stability of Brunn-Minkowski inequalities and Minkowski type problems for nonlinear capacity

位势理论的起源可以追溯到1687年牛顿关于力学定律的工作，当时他研究了遵循引力定律的力的性质。这一理论在17世纪和18世纪被拉格朗日、勒让德、拉普拉斯和高斯广泛用于研究引力、静电学和磁学理论中的问题。据观察，这些力可以用所谓的调和函数来模拟，调和函数是一种非常特殊的线性偏微分方程(PDE)的解，称为拉普拉斯方程。在物理学中出现了一个被称为容量的测量概念，它被定义为物体保持电荷的能力。在数学上，它可以用某个调和函数的积分来计算。电容在调和函数的研究中得到了广泛的应用，这一数学领域被称为位势理论。这一理论在很多方面都有分支，包括p-Laplace方程的非线性势理论和A-调和偏微分方程组。这些方程是二阶椭圆型偏微分方程组，可以看作是拉普拉斯方程的非线性推广。由于a-谐波偏微分方程的非线性，近年来在流变学、冰川学、热辐射、塑料成型等领域得到了较少的关注。在研究A-调和偏微分方程边值问题时，自然会出现与A-调和偏微分方程组相关的非线性容量。在凸分析中会出现一种称为集合的Minkowski加法的数学运算。它是由集合中所有可能的和相加定义的，它出现在运动规划、三维实体建模、聚集理论和碰撞检测中。经典的Brunn-Minkowski不等式早在一个多世纪前就已为人所知，它将欧几里得空间的子集的体积与Minkowski加法联系起来。对于包括C.Borell所获得的容量在内的各种其他量，它都已获得。最近，PI和他的合作者观察到非线性容量满足Brunn-Minkowski类型的不等式，并指出它的某个幂是包含低维集合的任何凸紧集的Minkowski加法下的凹函数。受M.Christian，A.Figalli和D.Jerison关于经典Brunn-Minkowski不等式稳定性的最新发展的启发，本项目的第一部分致力于研究与凸紧集上的A-调和偏微分方程有关的非线性容量的Brunn-Minkowski不等式的稳定性。这是一个新的具有挑战性的研究方向，因为这个问题甚至对于与拉普拉斯有关的对数或牛顿容量都没有得到解决。该项目还将研究非凸集上这些不等式的锐性。一旦研究了Brunn-Minkowski不等式，自然就会研究一个相关的问题，即Minkowski问题。这个问题在于从数据中找到一个凸多面体，该数据由它们的面的法线和它们的表面面积组成。在光滑情况下，凸体的相应问题是求出给定其边界的高斯曲率的凸体，作为单位法线的函数。证明由三部分组成：存在性、唯一性和正则性。当基本方程为A-调和偏微分方程组时，PI和他的合作者从势论的角度研究了这一问题，并解决了这一问题的存在唯一性问题。该项目的第二部分集中于与A-调和偏微分方程组相关的非线性容量Minkowski问题的正则性。这就需要进一步研究一类包含Monge-Ampere方程、一类特殊的二阶非线性偏微分方程组和A-调和偏微分方程解的正则性。在D.Jerison工作的基础上，该项目还旨在通过研究Minkowski类型的问题来增加对与A-调和偏微分方程组相关的凸域的A-调和测度的理解。

The origin of potential theory goes back to Newton's work on laws of mechanics in 1687 while studying the properties of forces which follow the law of gravitation. This theory has been widely used during the 17th and 18th centuries by Lagrange, Legendre, Laplace, and Gauss to study problems in the theory of gravitation, electrostatics and magnetism. It was observed that these forces could be modeled using so called harmonic functions which are solutions to a very special linear partial differential equation (PDE) known as Laplace's equation. A measuring notion called capacity appears in Physics and is defined as the ability of a body to hold an electrical charge. Mathematically, it can be calculated in terms of an integral of a certain harmonic function. The capacity has been widely used while studying harmonic functions and this field of Mathematics is called Potential Theory. This theory branched off in many directions including nonlinear potential theory of p-Laplace equation and A-harmonic PDEs. These are second-order elliptic PDEs and can be seen as a nonlinear generalization of Laplace's equation. A-harmonic PDEs have received little attention due to their nonlinearity and recently found applications in rheology, glaciology, radiation of heat, plastic moulding. Nonlinear capacity associated to A-harmonic PDEs naturally appears while studying boundary value problems for A-harmonic PDEs.A mathematical operation called Minkowski addition of sets appears in convex analysis. It is defined by addition of all possible sums in the sets and it appears in motion planning, 3D solid modeling, aggregation theory, and collision detection. Classical Brunn-Minkowski inequality has been known for more than a century and relates the volumes of subsets of Euclidean space under the Minkowski addition. It has been obtained for various other quantities including capacity obtained by C. Borell. Recently, the PI and his collaborators observed that nonlinear capacity satisfies a Brunn-Minkowski type inequality and it states that a certain power of it is a concave function under the Minkowski addition of any convex compact sets including low-dimensional sets. Inspired by the recent development on stability of the classical Brunn-Minkowski inequality by M. Christ, A. Figalli, and D. Jerison, the first part of this project is devoted to studying the stability of Brunn-Minkowski inequality for nonlinear capacity associated to A-harmonic PDEs for convex compact sets. This is a new and challenging direction of research as this problem has not been addressed even for the Logarithmic or Newtonian capacity associated to Laplacian. The project will also investigate sharpness of these inequalities for non-convex sets. Once the Brunn-Minkowski inequality has been studied, it is natural to study a related problem which is known as the Minkowski problem. This problem consists in finding a convex polyhedron from data consisting of normals to their faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary, as a function of the unit normal. The proof consists of three parts: existence, uniqueness, and regularity. The PI and his collaborators have studied this problem from the potential theoretic point of view when underlying equations are A-harmonic PDEs and solved the existence and uniqueness in this setting. The second part of the project focuses on regularity of the Minkowski problem for nonlinear capacity associated to A-harmonic PDEs. This requires further work on regularity of solutions to a system of PDEs involving Monge-Ampere equation, a nonlinear second-order PDE of special kind, and A-harmonic PDEs. Building on D. Jerison's work, the project also aims to increase understanding of A-harmonic measures of convex domains associated to A-harmonic PDEs by studying a Minkowski-type problem.