Qualitative Study of the Mean Field Equation and Allen-Cahn Equation

平均场方程和Allen-Cahn方程是在研究电弱理论和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时产生的两类重要的非线性偏微分方程组，平均场方程还与广义相对论研究中的Hawking质量刚性有关，以及与爱因斯坦理论耦合的大质量W玻色子模型的自引力弦有关。艾伦-卡恩方程的一个重要方面是显示分隔不同物理感兴趣区域的界面。这样的界面通常有一个在肥皂泡或数学术语中的最小表面中看到的重要特征。该方程还在许多其他科学和工程领域中得到了应用，如天体物理和图像处理。这些方程式也为训练学生和初级研究人员在涉及科学和工程数学方法的跨学科研究方面提供了极好的模式。国际和平研究所建议从事这些方程式的研究和培训，并让本科生和研究生参与跨学科研究。博士后研究员和初级研究人员也将参与并接受该项目的培训。PI计划研究PI和他的合作者最近发现的球面覆盖不等式(SCI)，包括它的推广和应用，特别是平均场方程及其类型。SCI将几何与分析联系起来，已成为研究非线性偏微分方程组中二维问题的有力工具。PI打算将其扩展到更高的维度。对于Allen-Cahn方程，PI将关注具有给定水平集的特解的存在性以及有限Morse指标解的水平集结构，特别是解的水平集与极小曲面之间的关系。PI打算利用各种恒等式以及Morse指数信息来开发这些非单调、非最小化解的新方法。长期目标是完全理解标量和矢量值Allen-Cahn方程的完整解，以及三重结或四重结的稳定性和动力学。Allen-Cahn方程解的节点集或奇点将在研究中受到特别关注，因为它们不仅在方程的理论分析中发挥着重要作用，而且在应用中也代表了晶体合金等材料中不同相或晶界的界面或界面的连接。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。

The mean field equation and the Allen-Cahn equation are two important types of nonlinear partial differential equations (PDEs) which have arisen in the study of several physical phenomena such as Electroweak theory and Chern-Simons-Higgs quantum field theories, statistical mechanics of two-dimensional turbulence, phase separation and phase transition, etc. The mean field equation is also related to the rigidity of Hawking Mass in the study of general relativity as well as to self-gravitating strings for a massive W-boson model coupled to Einstein theory in account of gravitational effects in cosmology. An important aspect of the Allen-Cahn equation is the display of interfaces separating different physical regions of interests. Such interfaces often share a significant feature seen in soap bubbles, or minimal surfaces in mathematical terminology. The equation has also found applications in many other area of sciences and engineering such as astrophysics and image processing. These equations also provide excellent models for training students and junior researchers in interdisciplinary research involving mathematical methods for sciences and engineering. The PI proposes to engage in both research and training aspects of these equations and to involve students at both undergraduate level and graduate level in interdisciplinary research. Postdoctoral fellows and junior researchers will also participate and be trained in the project. The PI plans to investigate the Sphere Covering Inequality (SCI) recently discovered by the PI and his collaborator, including its generalizations and applications, in particular to the mean field equation and its type. The SCI connects geometry to analysis and has become a powerful tool in the study of two dimensional problems in nonlinear PDEs. The PI intends to extend it to high dimensions. For the Allen-Cahn equation, the PI will focus on the existence of special solutions with prescribed level sets as well as on the level set structure of solutions of finite Morse index, in particular, on the relation between the level sets of solutions and minimal surfaces. The PI intends to use various identities as well as Morse index information to develop new approach for these non monotone, non minimizing solutions. The long term goal is to understand completely entire solutions for both scalar and vector-valued Allen-Cahn equations and the stability and dynamics of triple junctions or quadruple junctions. The nodal sets or singularities of the solutions of Allen-Cahn equation will receive special attention in the study since they not only play an important role in the theoretic analysis of the equation, they also represent in applications the interfaces or junctions of interfaces of different phases or grain boundaries in materials such as crystalline alloys.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.