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Qualitative Properties of Solutions to Nonlinear Elliptic Partial Differential Equations

非线性椭圆偏微分方程解的定性性质

基本信息

批准号：
1800645
负责人：
Ovidiu Savin
金额：
$ 25.17万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800645&HistoricalAwards=false
关键词：
Qualitative Properties Solutions Nonlinear Elliptic

项目摘要

This project focuses on central problems in classical fields of analysis such as the calculus of variations and free boundary problems. The problems under investigation share some common features and have roots in different areas like material sciences, fluid mechanics, geometry, cost optimization etc. They are relevant to both pure and applied mathematics. The proposed problems require progress on the known methods and techniques and addressing them would be beneficial for mathematicians and possibly for the larger scientific community as well. The outcome will be disseminated to the interested audience to invigorate the advancement of the theory. A central part of the project deals with the stability of singular minimizing maps of smooth, convex energies that appear in nonlinear elasticity. The PI will investigate the singularity formation in the associated parabolic flow and the regularity of solutions to a class of elliptic systems with special structure. Another part of the project is concerned with problems in nonlinear elliptic equations. The PI shall study the boundary behavior of some degenerate Monge-Ampere equations and optimal transportation problems, and to develop Pogorelov type estimates for some interesting model equation. In free boundaries, the PI will study the two-phase free boundary problem for different operators together with various obstacle-type problems, and will also analyze the rigidity of boundary layer phase transitions.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.

本项目关注经典分析领域的核心问题，如变分法和自由边界问题。所研究的问题具有一些共同的特点，并且植根于材料科学、流体力学、几何、成本优化等不同领域。它们与纯数学和应用数学都有关系。提出的问题需要在已知的方法和技术上取得进展，解决这些问题将有利于数学家，也可能有利于更大的科学界。结果将传播给感兴趣的观众，以激发理论的进步。该项目的核心部分是处理非线性弹性中出现的光滑凸能量的奇异最小映射的稳定性。PI将研究相关抛物流中的奇点形成和一类特殊结构椭圆系统解的正则性。项目的另一部分涉及非线性椭圆方程的问题。PI将研究一些退化的蒙日-安培方程和最优运输问题的边界行为，并对一些有趣的模型方程进行Pogorelov型估计。在自由边界中，PI将研究不同算符的两相自由边界问题以及各种障碍型问题，并分析边界层相变的刚性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。