Regularity Problems in the Calculus of Variations and Elliptic Partial Differential Equations

变分和椭圆偏微分方程中的正则问题

• 批准号: 1500438
• 负责人: Ovidiu Savin
• 金额: $ 22.59万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500438&HistoricalAwards=false
• 关键词: Regularity; Problems; Calculus; Variations; Elliptic

This project focuses on central problems in classical fields such as the calculus of variations and free boundary problems. Some problems find their motivation in the applied sciences and therefore offer opportunities for collaborations among mathematicians and other researchers. For example, the so-called "one-phase problem" describes the motion of a fluid in which a cavity is present that is a mixture of vapor and gas and the pressure on the cavity is constant. Other problems under investigation are related to issues of optimal transportation and allocation of resources. All such problems require the development of new sophisticated techniques, and progress will be disseminated to the scientific community to invigorate the advancement of the theory.The PI will study the classification of cones in low dimensions for the "one-phase" free boundary problem. As a tool in the analysis of free boundaries, the PI will investigate some boundary Harnack type results. In the context of optimal transportation, the PI is interested in the higher regularity for a class of degenerate Monge-Ampere equations in which the right hand side vanishes on the boundary. Finally, parts of this project deal with the fundamental question of local regularity of minimizers in the calculus of variations. In particular, in the scalar case the PI will investigate situations in which the integrand becomes highly degenerate on some compact set. In the vectorial case, the PI plans to construct some interesting singular minimizers in low dimensions and will also explore a viscosity approach to regularity for certain types of functionals.

本课题主要研究变分法和自由边界问题等经典领域的核心问题。一些问题在应用科学中找到了它们的动机，因此为数学家和其他研究人员之间的合作提供了机会。例如，所谓的“单相问题”描述了一种流体的运动，其中存在一个由蒸汽和气体混合而成的空腔，并且空腔上的压力是恒定的。正在调查的其他问题与最佳运输和资源分配问题有关。所有这些问题都需要发展新的复杂技术，并将其进展传播给科学界，以促进理论的发展。PI将研究“单相”自由边界问题的低维锥的分类。作为自由边界分析的工具，PI将研究一些边界哈纳克型结果。在最优输运的情况下，PI对一类退化蒙日-安培方程的高正则性感兴趣，其中右侧在边界上消失。最后，本项目的部分内容涉及变分法中最小值的局部正则性的基本问题。特别地，在标量情况下，PI将研究被积函数在紧集上高度简并的情况。在向量的情况下，PI计划在低维中构造一些有趣的奇异最小化器，并且还将探索一种粘性方法来研究某些类型的泛函的正则性。