CAREER: Existence, regularity, uniqueness and stability in anisotropic geometric variational problems

最小化面积泛函是数学中最著名的几何变分问题之一，它在数学和物理学中都产生了重大影响。然而，对于一些自然现象，面积泛函只是第一近似。为了捕捉微观结构，科学中的大多数模型使用方向相关泛函，称为各向异性能量。自世纪吉布斯将各向异性能引入晶体模型以来，它们已广泛应用于材料科学和工程，并在几何分析中产生了开创性的工作。然而，由于各向异性能量在平移和旋转下不是不变的，它们不享受面积泛函的守恒定律，这使得它们的研究变得更加复杂。PI将推进各向异性极小曲面理论。理解各向异性几何变分问题解的存在性、正则性、唯一性和稳定性在分析、几何、拓扑和物理中起着重要作用。PI还将开展与研究活动相关的垂直整合教育活动。特别是，本科生和研究生将通过组织研讨会，会议和暑期学校接触到这个项目的问题和技术。PI将研究各向异性极小曲面的存在性，正则性，唯一性和稳定性。黎曼流形中各向异性极小曲面的存在性需要推广极大极小理论。为了确定各向异性极小曲面的正则性，PI将研究相关的几何非线性椭圆偏微分方程。除了固定的配置，这项研究将揭示各向异性的Brakke流及其近似，通过相关的抛物偏微分方程的分析。本计画亦将探讨各向异性等周问题临界点的唯一性，并研究武尔夫形状的稳定性。此外，该研究的一部分将致力于最佳运输，重点是分支动力学的规律性和稳定性。该奖项反映了NSF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。

Minimizing the area functional is one of the most famous examples of geometric variational problems in mathematics and it has had a major impact both in mathematics and in physics. However, for several natural phenomena, the area functional is only a first approximation. In order to capture microstructures, most models in the sciences use directionally dependent functionals, referred to as anisotropic energies. Ever since the introduction of anisotropic energies by Gibbs in the 19th century to model crystals, they have been extensively applied in material sciences and engineering, motivating seminal works in geometric analysis. However, since anisotropic energies are not invariant under translations and rotations, they don’t enjoy the conservation laws of the area functional, which makes them significantly more complicated to study. The PI will advance the anisotropic minimal surfaces theory. Understanding existence, regularity, uniqueness and stability of solutions to anisotropic geometric variational problems plays a major role in analysis, geometry, topology and physics. The PI will also conduct vertically integrated educational activities tied with the research activities. In particular, undergraduate and graduate students will be exposed to the problems and techniques of this project via the organization of seminars, conferences, and a summer school.The PI will study existence, regularity, uniqueness and stability properties of anisotropic minimal surfaces. The existence of anisotropic minimal surfaces in Riemannian manifolds will require extending the min-max theory. In order to determine the regularity of anisotropic minimal surfaces, the PI will study the related geometric nonlinear elliptic PDEs. In addition to the stationary configurations, this research will shed light on the anisotropic Brakke flow and its approximation, through the analysis of the related parabolic PDEs. This project will also address the uniqueness of critical points of the anisotropic isoperimetric problem and investigate the stability properties of the Wulff shapes. Furthermore, part of this research will be devoted to optimal transport, with an emphasis on the regularity and stability properties of branching dynamics.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.