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。这些问题具有一些共同的特征，并且由各种物理现象激发，例如彼此接触的弹性膜的相互作用，流体的射流，最小面积的表面以及两个区域的元件之间的最佳运输。这些问题的理论知识的进步将有利于科学界的一般，并可能有应用程序，以更具体的计算方面，解决这些方程的数值。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特殊自由边界问题的正则性理论和非线性偏微分方程。第一部分是关于特殊拉格朗日方程中奇异性的形成。该方程出现在校准几何和极小子流形的上下文中。主要研究员（PI）研究小扰动下奇异解的稳定性以及与其研究相关的某些退化Bellman方程。第二部分是自由边界问题。PI研究在两相Alt-Phillips自由边界问题研究中出现的正则性问题。一些相关的问题涉及刚性的整体解决方案，在低维的德Giorgi猜想的精神。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。该项目的另一部分是关于非局部最小图的规律性和非局部算子的边界Harnack原理的一些相关问题。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.