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Interacting Free Boundaries in the Calculus of Variations

变分法中相互作用的自由边界

基本信息

批准号：
2055617
负责人：
Ovidiu Savin
金额：
$ 25.45万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055617&HistoricalAwards=false
关键词：
Interacting Free Boundaries Calculus Variations

项目摘要

The goal of the project is to develop further the mathematical tools needed in the study of fundamental questions in nonlinear partial differential equations, calculus of variations, and free boundary problems. These questions have their origins in the applied sciences like elasticity, fluid mechanics, or cost optimization, and are relevant in areas of pure mathematics like geometry. The theoretical aspects of these models are important towards a better understanding of basic physical phenomena and could be useful for the larger scientific community. The project provides research training opportunities for students and its outcomes will be broadly disseminated to diverse audiences. The principal investigator (PI) will study several questions related to the classical obstacle problem. One of them concerns the multiple membrane problem, which describes the interaction of N elastic membranes in contact with each other. Mathematically, it can be viewed as a coupled system of obstacle problems with N-1 interacting free boundaries. Another question to be addressed deals with the uniqueness of certain blow-up cones for the thin obstacle problem, also known as the Signorini problem. An important part of the project considers the optimal transportation problem for quadratic costs with densities that vanish on the boundary of their support. This problem is equivalent to the second boundary value problem for a highly degenerate Monge-Ampère equation. The PI also aims to understand the interior regularity of minimizing maps in the calculus of variations for one of the simplest polyconvex energies in two dimensions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的目标是进一步开发研究非线性偏微分方程、变分法和自由边界问题等基本问题所需的数学工具。这些问题起源于弹性、流体力学或成本优化等应用科学，并且与几何等纯数学领域相关。这些模型的理论方面对于更好地理解基本物理现象非常重要，并且可能对更大的科学界有用。该项目为学生提供研究培训机会，其成果将广泛传播给不同的受众。首席研究员（PI）将研究与经典障碍问题相关的几个问题。其中之一涉及多膜问题，该问题描述了相互接触的 N 个弹性膜的相互作用。从数学上讲，它可以被视为具有 N-1 个相互作用的自由边界的障碍问题的耦合系统。另一个需要解决的问题涉及薄障碍问题（也称为西格诺里尼问题）的某些爆炸锥体的独特性。该项目的一个重要部分考虑了二次成本的最优运输问题，其密度在其支持边界上消失。该问题相当于高度简并 Monge-Ampère 方程的第二个边值问题。该 PI 还旨在了解二维最简单的多凸能量之一的变分微积分中最小化图的内部规律。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。