Unique continuation and the regularity of elliptic PDEs and generalized minimal submanifolds

椭圆偏微分方程和广义最小子流形的唯一延拓和正则性

• 批准号: 2350351
• 负责人: Zihui Zhao
• 金额: $ 25.37万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350351&HistoricalAwards=false
• 关键词: Unique; continuation; regularity; elliptic; PDEs

This award supports research on the regularity of solutions to elliptic partial differential equations and regularity of generalized minimal submanifolds. Elliptic differential equations govern the equilibrium configurations of various physical phenomena, for instance, those arising from minimization problems for natural energy functionals. Examples include the shape of free-hanging bridges, the shape of soap bubbles, and the sound of drums. Elliptic differential equations are also used to quantify the degree to which physical objects are bent or distorted, with far-reaching implications and applications in geometry and topology. The proposed research focuses on the regularity of solutions to such equations. Questions to be addressed include the following: Do non-smooth points (singularities) exist? How large can the set of singularities be? What is the behavior of the solution near a singularity? Is it possible to perturb the underlying environment in order to eliminate the singularity? The project will also provide opportunities for the professional development of graduate students, both via individual mentoring and via the organization of a directed learning seminar on geometric analysis and geometric measure theory.The mathematical objectives of the project are twofold. First, the principal investigator will study unique continuation for solutions to elliptic partial differential equations, with a focus on quantitative estimates on the size and structure of the singular set of these solutions. A second topic for consideration is the regularity theory for generalized minimal submanifolds (a generalized notion of smooth submanifolds which arise as critical points for the area functional under local deformations). In particular, the principal investigator will study branch singular points in the interior as well as at the boundary of a generalized minimal submanifold, under an area-minimizing or stability assumption. Research on the latter topic, which can be viewed as a non-linear analogue of quantitative unique continuation for elliptic equations, requires the integration of ideas from geometric measure theory, partial differential equations and geometric analysis.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.

该奖项支持对椭圆偏微分方程解的正则性和广义最小子流形正则性的研究。椭圆微分方程控制各种物理现象的平衡配置，例如，自然能量泛函的最小化问题引起的平衡配置。例如，自由悬挂的桥梁的形状、肥皂泡的形状和鼓的声音。椭圆微分方程还用于量化物理对象弯曲或扭曲的程度，在几何和拓扑中具有深远的影响和应用。拟议的研究重点是此类方程的解的规律性。要解决的问题包括以下内容：是否存在非光滑点（奇点）？奇点集可以有多大？奇点附近的解的行为是什么？是否有可能扰动底层环境以消除奇点？该项目还将通过个人指导和组织几何分析和几何测量理论的定向学习研讨会，为研究生的专业发展提供机会。该项目的数学目标是双重的。首先，首席研究员将研究椭圆偏微分方程解的唯一延拓，重点是对这些解的奇异集的大小和结构的定量估计。第二个需要考虑的主题是广义最小子流形的正则理论（平滑子流形的广义概念，作为局部变形下面积泛函的临界点而出现）。特别是，主要研究者将在面积最小化或稳定性假设下研究广义最小子流形内部和边界的分支奇点。后一个主题的研究可以看作是椭圆方程定量唯一延拓的非线性模拟，需要整合几何测度论、偏微分方程和几何分析的思想。该奖项反映了 NSF 的法定使命，并已被通过使用基金会的智力优点和更广泛的影响审查标准进行评估，认为值得支持。