Regularity Problems in Free Boundaries and Degenerate Elliptic Partial Differential Equations

自由边界和简并椭圆偏微分方程中的正则问题

• 批准号: 2349794
• 负责人: Ovidiu Savin
• 金额: $ 27.39万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349794&HistoricalAwards=false
• 关键词: Regularity; Problems; Free; Boundaries; Degenerate

The goal of this project is to develop new methods for the mathematical theory in several problems of interest involving partial differential equations (PDE). The problems share some common features and are motivated by various physical phenomena such as the interaction of elastic membranes in contact with one another, jet flows of fluids, surfaces of minimal area, and optimal transportation between the elements of two regions. Advancement in the theoretical knowledge about these problems would be beneficial to the scientific community in general and possibly have applications to more concrete computational aspects of solving these equations numerically. The outcomes of the project will be disseminated at a variety of seminars and conferences. The project focuses on the regularity theory of some specific free boundary problems and nonlinear PDE. The first part is concerned with singularity formation in the Special Lagrangian equation. The equation appears in the context of calibrated geometries and minimal submanifolds. The Principal Investigator (PI) studies the stability of singular solutions under small perturbations together with certain degenerate Bellman equations that are relevant to their study. The second part of the project is devoted to free boundary problems. The PI investigates regularity questions that arise in the study of the two-phase Alt-Phillips family of free boundary problems. Some related questions concern rigidity of global solutions in low dimensions in the spirit of the De Giorgi conjecture. A second problem of interest involves coupled systems of interacting free boundaries. They arise in physical models that describe the configuration of multiple elastic membranes that are interacting with each other according to some specific potential. Another part of the project is concerned with the regularity of nonlocal minimal graphs and some related questions about the boundary Harnack principle for nonlocal operators.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.

本课题的目标是为涉及偏微分方程（PDE）的几个有趣问题的数学理论开发新的方法。这些问题有一些共同的特点，并且是由各种物理现象引起的，如弹性膜相互接触的相互作用、流体的射流、最小面积表面和两个区域元素之间的最佳运输。关于这些问题的理论知识的进步将有利于科学界，并可能应用于数值解这些方程的更具体的计算方面。该项目的成果将在各种研讨会和会议上传播。本课题主要研究一些特定自由边界问题的正则性理论和非线性偏微分方程。第一部分讨论了特殊拉格朗日方程奇点的形成。该方程出现在校准几何和最小子流形的背景下。主要研究者（PI）研究了在小扰动下奇异解的稳定性，并给出了与其研究相关的退化Bellman方程。项目的第二部分致力于自由边界问题。PI研究了在研究两相Alt-Phillips族自由边界问题时出现的正则性问题。一些相关的问题涉及低维全局解的刚性，这与德乔治猜想的精神有关。第二个令人感兴趣的问题涉及相互作用的自由边界的耦合系统。它们出现在描述多个弹性膜的结构的物理模型中，这些弹性膜根据某些特定的势相互作用。另一部分研究了非局部极小图的正则性和非局部算子的边界哈纳克原理的相关问题。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。