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.

潜在理论的起源可以追溯到牛顿在1687年研究力学定律的工作，同时研究了遵循重力定律的力的特性。在17和18世纪，Lagrange，Legendre，Laplace和Gauss在17和18世纪广泛使用了该理论，以研究引力，静电和磁性理论的问题。据观察，这些力可以使用所谓的谐波函数对这些力进行建模，这是一种非常特殊的线性部分微分方程（PDE）的解决方案，称为Laplace的方程。一个称为容量的测量概念出现在物理学中，并定义为身体容纳电荷的能力。从数学上讲，它可以根据某个谐波函数的积分来计算。在研究谐波功能时，能力已被广泛使用，并且这种数学领域称为潜在理论。该理论在许多方向上进行了分支，包括p-Laplace方程和A谐波PDE的非线性潜在理论。这些是二阶椭圆PDE，可以看作是Laplace方程的非线性概括。 A-Harmonic PDE由于其非线性而受到很少的关注，最近在流变学，冰川学，热的辐射，塑料成型中发现了应用。在研究A Harmonic PDE的边界值问题时，自然出现了与A Harmonic PDES相关的非线性容量。一种称为Minkowski添加集的数学操作出现在CONVEX分析中。它是通过在集合中添加所有可能的总和来定义的，并且在运动计划，3D固体建模，聚合理论和碰撞检测中出现。经典的布鲁恩 - 米科夫斯基（Brunn-Minkowski）不平等现象已有一个多世纪的历史，并在Minkowski的增加下将欧几里得空间的子集卷汇总。它是针对C. Borell获得的各种其他数量的各种数量获得的。最近，PI和他的合作者观察到，非线性容量满足了Brunn-Minkowski型不平等，并指出，在Minkowski添加任何凸的紧凑型集合（包括低维集）的情况下，IT的某些功能是凹功能。受到M. Christ，A。Figalli和D. Jerison的经典Brunn-Minkowski不平等的稳定性发展的启发，该项目的第一部分致力于研究Brunn-Minkowski不平等的稳定性，即与A-HarmoneC PDE相关的非线性能力的稳定性。这是一个新的挑战性研究方向，因为即使对于与拉普拉斯（Laplacian）相关的对数或牛顿的能力，这个问题也没有解决。该项目还将调查这些不平等现象的清晰度。一旦研究了Brunn-Minkowski的不平等现象，就可以研究一个被称为Minkowski问题的相关问题。这个问题包括从一个由正常的数据组成的数据到其面部和表面区域的数据中找到凸多面体。在平滑的情况下，凸体的相应问题是在其边界的高斯曲率下找到凸体，这是单位正常的函数。证明包括三个部分：存在，独特性和规律性。 PI和他的合作者从潜在的理论观点研究了这个问题时，底层方程式为a谐波PDE，并在这种情况下解决了存在和唯一性。该项目的第二部分重点是与A谐波PDE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Monge-Ampere方程，非线性二阶PDE的PDE系统的规律性，特殊类型的PDE和A-Harmonic PDE。在D. Jerison的工作的基础上，该项目还旨在通过研究Minkowski-type问题来提高对与A谐波PDE相关的凸形域的A谐波测量的理解。