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.

该项目的目的是在涉及部分微分方程（PDE）的几个感兴趣的问题中开发数学理论的新方法。这些问题具有一些共同的特征，并由各种物理现象（例如彼此接触的弹性膜的相互作用），流体的喷射流，最小面积的表面以及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对一般的科学界有益，并且可能在数值上求解这些方程式的更具体的计算方面。该项目的结果将在各种研讨会和会议上传播。 该项目着重于一些特定的自由边界问题和非线性PDE的规律性理论。第一部分与特殊拉格朗日方程中的奇异性形成有关。该方程式出现在校准的几何形状和最小的亚策略的背景下。首席研究者（PI）研究了在小扰动下奇异溶液的稳定性以及与其研究相关的某些退化钟形方程。该项目的第二部分致力于自由边界问题。 PI调查了对两相替补阶段的自由边界问题家族的研究时期出现的规律性问题。一些相关问题涉及全球解决方案在低维度的僵化，以de giorgi的猜想精神。感兴趣的第二个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配置的物理模型中，这些弹性膜根据某些特定的潜力相互相互作用。该项目的另一部分涉及非局部最小图的规律性以及有关非本地运营商边界Harnack原理的一些相关问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准通过评估来进行评估的。