Studies of the Mean Field and Allen-Cahn Equations

偏微分方程（PDE）是模拟许多科学和社会现象的基本工具。平均场方程和Allen-Cahn方程是两种重要的非线性偏微分方程，它们在研究诸如电弱相互作用、Chern-Simons-Higgs量子场论、二维湍流统计力学、相分离和相变等物理现象中出现。平均场方程也与广义相对论和宇宙学的研究有关。艾伦-卡恩方程的一个重要方面是形成分离不同物理区域的界面，这些界面通常具有面积最小化特性，如肥皂泡等最小表面。该研究项目旨在通过使用研究人员和合作者最近开发的数学工具来促进对这些重要方程的理解。该项目涉及跨学科研究的本科生和研究生。博士后研究员和初级研究人员也将参与并接受培训，作为项目的一部分。首席研究员计划研究新发现的领域，涵盖高维不平等（SCI）和各种其他推广。将探讨平均场方程及其高维对应方程的应用。SCI将几何与分析联系起来，已成为研究二维非线性偏微分方程的有力工具。高维SCIs在共形几何的研究中有很大的潜力，如改进的Beckner不等式和含Paneitz算子的高阶方程解的唯一性。对于Allen-Cahn方程，PI将关注有限莫尔斯指数的解的水平集结构，例如，解的水平集与极小曲面之间的关系。特别是，将在三个方面对稳定的解决办法进行分类。PI打算使用各种身份和能量估计以及莫尔斯指数信息来开发这些非单调，非最小化解决方案的新方法。该项目的最终目标是完全理解标量和矢量值Allen-Cahn方程的完整解以及三重或四重结的稳定性和动力学。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而被认为值得支持。

Partial differential equations (PDE) are fundamental tools in modeling many scientific and social phenomena. The mean field equation and the Allen-Cahn equation are two important types of nonlinear PDE that have arisen in the study of physical phenomena such as electroweak interaction and Chern-Simons-Higgs quantum field theories, statistical mechanics of two-dimensional turbulence, phase separation, and phase transitions. The mean field equation is also related to the study of general relativity as well as to cosmology. An important aspect of the Allen-Cahn equation is the formation of interfaces separating different physical regions of interest, which often have area minimizing properties, seen in minimal surfaces such as soap bubbles. This research project aims to advance understanding of these important equations through use of a mathematical tool recently developed by the investigator and collaborators. The project involves students at both undergraduate and graduate levels in interdisciplinary research. Postdoctoral fellows and junior researchers will also participate in and be trained as part of the project.The principal investigator plans to investigate the newly discovered sphere covering inequality (SCI) in high dimensions and various other generalizations. Applications to the mean field equation and its high-dimensional counterpart will be explored. The SCI connects geometry to analysis and has become a powerful tool in the study of two-dimensional nonlinear PDE. High-dimensional SCIs have potential for study of conformal geometry, such as improved Beckner's inequalities and uniqueness of solutions to higher order equations involving Paneitz operators. For the Allen-Cahn equation, the PI will focus on the level set structure of solutions of finite Morse index, such as, for example, the relation between the level sets of solutions and minimal surfaces. In particular, a classification of stable solutions in three dimensions will be pursued. The PI intends to use various identities and energy estimates as well as Morse index information to develop a new approach for these non-monotone, non-minimizing solutions. The final goal of the project is to understand completely entire solutions for both scalar and vector-valued Allen-Cahn equations and the stability and dynamics of triple or quadruple junctions.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.