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

Brunn-Minkowski 不等式的稳定性和非线性容量的 Minkowski 型问题

基本信息

批准号：
EP/W001586/1
负责人：
Murat Akman
金额：
$ 32.14万
依托单位：
University of Essex
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW001586%2F1
关键词：
Stability Brunn Minkowski inequalities type

项目摘要

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.

势能理论的起源可以追溯到 1687 年牛顿对力学定律的研究，当时他研究了遵循万有引力定律的力的性质。该理论在 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来研究引力、静电和磁学理论中的问题。据观察，这些力可以使用所谓的调和函数来建模，调和函数是一种非常特殊的线性偏微分方程（PDE）（称为拉普拉斯方程）的解。物理学中出现了一个称为容量的测量概念，定义为物体保持电荷的能力。在数学上，它可以根据某个调和函数的积分来计算。这种容量在研究调和函数时得到了广泛的应用，这个数学领域被称为势论。该理论向多个方向分支，包括 p-拉普拉斯方程的非线性势理论和 A 谐波偏微分方程。这些是二阶椭圆偏微分方程，可以看作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而很少受到关注，最近在流变学、冰川学、热辐射、塑料成型中得到了应用。在研究 A 调和偏微分方程的边值问题时，自然会出现与 A 调和偏微分方程相关的非线性容量。凸分析中出现了一种称为明可夫斯基集合加法的数学运算。它是通过将集合中所有可能的和相加来定义的，并且出现在运动规划、3D 实体建模、聚合理论和碰撞检测中。经典的 Brunn-Minkowski 不等式已为人所知一个多世纪了，它与 Minkowski 加法下的欧几里得空间子集的体积相关。它已获得各种其他数量，包括 C. Borell 获得的容量。最近，PI和他的合作者观察到非线性能力满足Brunn-Minkowski型不等式，并且指出它的某个幂是在任何凸紧集（包括低维集）的Minkowski加法下的凹函数。受 M. Christ、A. Figalli 和 D. Jerison 最近对经典 Brunn-Minkowski 不等式的稳定性研究的启发，该项目的第一部分致力于研究与凸紧集 A 调和 PDE 相关的非线性能力的 Brunn-Minkowski 不等式的稳定性。这是一个新的且具有挑战性的研究方向，因为即使对于与拉普拉斯相关的对数或牛顿容量，这个问题也尚未得到解决。该项目还将研究非凸集的这些不等式的锐度。一旦研究了 Brunn-Minkowski 不等式，就很自然地要研究一个相关的问题，即 Minkowski 问题。该问题在于从由面法线和表面积组成的数据中找到凸多面体。在光滑的情况下，凸体的相应问题是在给定凸体边界高斯曲率（作为单位法线的函数）的情况下找到凸体。证明由三部分组成：存在性、唯一性、规律性。 PI 和他的合作者从势理论的角度研究了当基础方程是 A 调和偏微分方程时的这个问题，并解决了这种情况下的存在性和唯一性。该项目的第二部分重点关注与 A 谐波偏微分方程相关的非线性容量的 Minkowski 问题的规律性。这需要进一步研究涉及蒙日-安培方程、特殊类型的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组的解的规律性。该项目以 D. Jerison 的工作为基础，还旨在通过研究 Minkowski 型问题来加深对与 A 调和偏微分方程相关的凸域的 A 调和测量的理解。