Regularity Problems in Free Boundaries and Degenerate Elliptic Partial Differential Equations

这个项目的目标是为涉及偏微分方程(PDE)的几个感兴趣的问题的数学理论开发新的方法。这些问题有一些共同的特征，并受到各种物理现象的驱动，例如相互接触的弹性膜的相互作用，流体的喷射流动，最小面积的表面，以及两个区域元素之间的最佳传输。关于这些问题的理论知识的进步总体上将有益于科学界，并可能应用于数值求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。本项目主要研究一些具体的自由边界问题和非线性偏微分方程组的正则性理论。第一部分是关于特殊拉格朗日方程的奇性形成。该方程出现在已校准几何和极小子流形的背景下。首席调查员(PI)研究奇异解在小扰动下的稳定性，以及与其研究相关的某些退化的Bellman方程。该项目的第二部分致力于自由边界问题。PI研究在研究两相Alt-Phillips族自由边界问题时出现的正则性问题。在德乔治猜想的精神下，一些相关的问题涉及低维全球解的刚性。第二个有趣的问题涉及相互作用的自由边界的耦合系统。它们出现在物理模型中，这些模型描述了根据特定势能相互作用的多个弹性膜的结构。该项目的另一部分涉及非局部极小图的正则性以及关于非局部算子的边界哈纳克原理的一些相关问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.