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

平均场方程和Allen-Cahn方程是两种重要的非线性偏微分方程，它们出现在电弱理论和chen - 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的研究中。平均场方程还与广义相对论研究中霍金质量的刚性以及与爱因斯坦宇宙学中引力效应理论耦合的大质量w -玻色子模型的自引力弦有关。Allen-Cahn方程的一个重要方面是显示分离不同物理兴趣区域的界面。这些界面通常具有肥皂泡或数学术语中的最小表面的重要特征。这个方程也在许多其他科学和工程领域得到了应用，比如天体物理学和图像处理。这些方程也为培养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中提供了很好的模型。PI建议对这些方程进行研究和训练，并让本科生和研究生都参与跨学科研究。博士后和初级研究人员也将参与并接受培训。PI计划调查PI和他的合作者最近发现的球体覆盖不等式（SCI），包括它的推广和应用，特别是在平均场方程及其类型上。SCI将几何与分析联系起来，成为研究非线性偏微分方程中二维问题的有力工具。PI打算将其扩展到高维。对于Allen-Cahn方程，PI将关注具有规定水平集的特解的存在性，以及有限Morse指数解的水平集结构，特别是解的水平集与极小曲面的关系。PI打算使用各种身份以及莫尔斯索引信息来开发这些非单调，非最小化解决方案的新方法。长期目标是完全理解标量和向量值Allen-Cahn方程的完整解，以及三联结或四联结的稳定性和动力学。Allen-Cahn方程解的节点集或奇点不仅在方程的理论分析中起着重要的作用，而且在应用中也代表了晶体合金等材料中不同相界面或晶界的界面或结。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。

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.