Analysis and Geometry of Free Boundaries

偏微分方程（PDE）是物理学的语言。事实上，第一个偏微分方程是在试图描述热的传播、波的传播以及电磁传播时自然产生的。工程中的许多问题也依赖于偏微分方程的理论，以及近似解的能力。该项目的科学部分涉及对模拟各种物理现象的特定偏微分方程族的研究。这些包括冰的融化、燃烧、化学扩散、液晶和图像处理。该项目的主要科学目标之一是开发新的数学工具，用于更好地理解正在建模的物理现象。这将创造新的途径来分析它们并提高它们的理解力。研究者还将组织一个为期一周的研讨会，重点关注对数学感兴趣的第一代本科生。学生们将参加小型课程，参加研究讲座，并与在不同领域工作的数学家进行非正式对话。该研讨会将有助于培养精通数学和多样化的劳动力。这个项目是由自由边界问题和几何测量理论中产生的问题驱动的。在应用科学中，人们经常遇到自由边界，当一个问题的解由一个函数（通常满足一个偏微分方程）和一个函数具有特定行为的集合组成时，就会出现自由边界。研究者将研究各种各样的问题，这些问题是由研究函数的规律性和相关集合的几何形状所激发的。这些是在理论和应用数学中普遍存在的核心问题，可以直接用于模拟各种物理现象。该项目的主要目标是有助于更好地理解涉及非局部方程的问题，具有自由边界的几乎最小化，以及各向异性能量的最小化。要研究的第一类问题涉及对数学建模具有基本重要性的偏微分方程。特别是，许多应用现象会产生非局部方程，如非局部图像处理和液晶。具有自由边界的几乎最小值的研究对于处理一组新的物理动机问题具有突出的潜力，因为几乎最小值的性质可以理解为具有噪声的最小化问题。最后，各向异性能量的最小化导致非均匀椭圆偏微分方程，在几何偏微分方程中产生新的、具有挑战性的问题。研究者将开发新的工具来解决自由边界问题和几何测量理论的相关问题。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。

Partial Differential Equations (PDE) are the language of physics. In fact, the first partial differential equations naturally arose in physics when trying to describe the propagation of heat, the propagation of waves as well as electromagnetism. Many problems in engineering also rely on the theory of PDEs, as well as the ability to approximate their solutions. The scientific part of this project is concerned with the study of specific families of PDEs that model various physical phenomena. These include the melting of ice, combustion, chemical diffusions, liquid crystals, and image processing. One of the main scientific goals of the project is to develop new mathematical tools that can be used to better understand the physical phenomena being modeled. This will create new avenues to analyze them and enhance their comprehension. The investigator will also organize a week-long workshop focused on first-generation undergraduate students who are interested in mathematics. The students will participate in minicourses, attend research talks, and have informal conversations with mathematicians who work in different sectors. The workshop will contribute to the development of a mathematically well-versed and diverse workforce. This project is driven by questions arising in free boundary problems and geometric measure theory. In the applied sciences one often encounters free boundaries, which arise when the solution to a problem consists of a function (often satisfying a partial differential equation) and a set where this function has a specific behavior. The investigator will study a variety of problems that are motivated by the study of the regularity of the function and the geometry of the associated set. These are central questions that are ubiquitous in both theoretical and applied mathematics and can be directly used to model various physical phenomena. The project’s main goal is to contribute to a better understanding of problems involving nonlocal equations, almost minimizers with free boundaries, and minimizers for anisotropic energies. The first class of problems to be investigated involves PDE of fundamental importance for mathematical modeling. In particular, numerous applied phenomena give rise to nonlocal equations, such as nonlocal image processing and liquid crystals. The study of almost minimizers with free boundaries has an outstanding potential to treat a new group of physically motivated problems, as the almost minimizing property can be understood as a minimizing problem with noise. Finally, minimizers for anisotropic energies lead to non-uniformly elliptic PDE, generating new, challenging questions in geometric PDE. The investigator will develop new tools which will address related questions at the interface of free boundary problems and geometric measure theory.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.