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年牛顿在研究万有引力定律下的力的性质时对力学定律的研究。这个理论在17世纪和18世纪被拉格朗日、勒让德、拉普拉斯和高斯广泛应用于研究引力、静电和磁力理论中的问题。人们观察到，这些力可以用所谓的调和函数来建模，调和函数是一个非常特殊的线性偏微分方程（PDE）的解，即拉普拉斯方程。物理学中出现了一种叫做容量的测量概念，它被定义为物体容纳电荷的能力。数学上，它可以用某个调和函数的积分来计算。在研究调和函数时，容量被广泛使用，这一数学领域被称为势论。该理论在p-拉普拉斯方程的非线性势理论和a -谐偏微分方程等方面得到了扩展。这些是二阶椭圆偏微分方程，可以看作是拉普拉斯方程的非线性推广。a -谐波偏微分方程由于其非线性特性而受到较少关注，最近在流变学、冰川学、热辐射、塑料成型等方面得到了应用。在研究a -谐波偏微分方程的边值问题时，自然会出现与之相关的非线性能力。在凸分析中出现了一种叫做闵可夫斯基集加法的数学运算。它是通过集合中所有可能的和的加法来定义的，它出现在运动规划，3D实体建模，聚合理论和碰撞检测中。经典的布伦-闵可夫斯基不等式已经被发现了一个多世纪，它与闵可夫斯基加法下欧几里德空间子集的体积有关。它也可用于其他各种量，包括由C. Borell得到的容量。最近，PI和他的合作者观察到非线性容量满足Brunn-Minkowski型不等式，并指出它的某个幂是在包括低维集在内的任何凸紧集的Minkowski加法下的凹函数。受M. Christ， A. Figalli和D. Jerison关于经典Brunn-Minkowski不等式稳定性的最新进展的启发，本课题的第一部分致力于研究凸紧集a -调和偏微分方程相关非线性容量的Brunn-Minkowski不等式的稳定性。这是一个新的和具有挑战性的研究方向，因为这个问题甚至没有解决与拉普拉斯相关的对数或牛顿能力。该项目还将研究非凸集的这些不等式的尖锐性。一旦研究了布伦-闵可夫斯基不等式，自然就会研究一个相关的问题，即闵可夫斯基问题。这个问题包括从它们的面和表面积的法线组成的数据中找到一个凸多面体。在光滑情况下，凸体的相应问题是给定凸体边界的高斯曲率作为单位法线的函数来求凸体。证明由存在性、唯一性和规律性三部分组成。PI和他的合作者从势理论的角度研究了底层方程为a -谐偏微分方程时的存在唯一性问题，并解决了这种情况下的存在唯一性问题。项目的第二部分重点研究了与a -谐波偏微分方程相关的非线性容量Minkowski问题的正则性。这需要进一步研究包含蒙日-安培方程、一类特殊非线性二阶偏微分方程和a -谐波偏微分方程的偏微分方程系统解的正则性。在d.j erison的工作基础上，该项目还旨在通过研究minkowski型问题来增加对与a -调和偏微分方程相关的凸域的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.