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

极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，面积泛函只是一次近似。为了捕捉微观结构，科学中的大多数模型使用方向相关的泛函，称为各向异性能量。自从19世纪Gibbs引入各向异性能量来模拟晶体以来，它们在材料科学和工程中得到了广泛的应用，激发了几何分析方面的开创性工作。然而，由于各向异性能量在平移和旋转下不是不变的，它们不符合面积泛函守恒定律，这使得它们的研究变得更加复杂。PI将推动各向异性极小曲面理论的发展。了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、拓扑学和物理学中具有重要意义。该协会还将开展与研究活动相联系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、会议和暑期班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、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需要推广极小极大理论。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型偏微分方程组。除了定常构型外，本研究还将通过对相关抛物型偏微分方程组的分析，阐明各向异性Brakke流及其近似。这个项目还将解决各向异性等周问题临界点的唯一性问题，并研究Wulff形状的稳定性。此外，这项研究的一部分将致力于最优运输，重点是分支动力的规律性和稳定性属性。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。

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.