Qualitative Studies of Some Partial Differential Equations and Systems

Allen-Cahn方程最早是由材料科学家提出的，用于研究不同材料之间的相分离以及同一材料不同相之间的相变。该方程的一个重要方面是，它允许显示分隔不同感兴趣的物理区域的界面。这样的界面通常与肥皂泡或更精确的数学术语--最小的表面--有显著的共同特征。该方程还在许多其他科学和工程领域得到了应用，如天体物理和图像处理。除了它在促进数学和其他科学知识方面的重要意义之外，艾伦-卡恩方程还为培训学生和初级研究人员进行跨学科研究提供了一个极好的工具。首席研究员将从事与方程相关的研究和培训活动，并将包括本科生和研究生，利用方程的理论研究来开发更好的医学图像分析算法。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。主要研究Allen-Cahn型方程的鞍点解和行波解，包括经典的双井势标量方程和多井势的矢量值方程。他将重点研究具有指定水平集的特殊鞍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。他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。该项目的长期目标是了解标量和矢量值Allen-Cahn方程的一般完整解，并深入了解三重结或四重结的稳定性和动力学。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起着重要作用，而且在应用中还代表了晶体合金等材料中不同相或晶界的界面或交界处。该项目将扩大代表不足的少数群体，包括西班牙裔学生对数学和跨学科研究的参与。

The Allen-Cahn equation was first proposed by materials scientists as a model for studying phase separation between different materials as well as phase transitions between different phases of the same materials. An important aspect of the equation is that it enables the display of interfaces separating different physical regions of interest. Such interfaces often share significant features with soap bubbles or, in more precise mathematical terminology, with minimal surfaces. The equation has also found applications in many other area of science and engineering such as astrophysics and image processing. In addition to its significance in advancing knowledge in mathematics and other sciences, the Allen-Cahn equation provides an excellent tool for training students and junior researchers in interdisciplinary research. The principal investigator will engage in both the research and the training activities related to the equation and will involve students at both the undergraduate and the graduate levels, using the theoretical study of the equation to develop better algorithms for medical image analysis. Postdoctoral fellows and junior researchers will also participate and be trained in the project. The principal investigator plans to study saddle solutions and traveling wave solutions of Allen-Cahn-type equations, including the classic scalar equations with double well potentials and vector-valued equations with multiple well potentials. He will focus on the existence of special saddle 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. He intends to use various identities, as well as Morse index information, to develop new approach for these nonmonotone, nonminimizing solutions. The long-term goal of the project is to understand general entire solutions to both scalar and vector-valued Allen-Cahn equations and to gain insight into the stability and dynamics of triple junctions or quadruple junctions. The nodal sets or singularities of the solutions will receive special attention in the study, for not only do they play an important role in the theoretical analysis of the equation, but they also represent in applications the interfaces or junctions of interfaces of different phases or grain boundaries in materials such as crystalline alloys. The project will broaden the participation of underrepresented minorities, including Hispanic students, in mathematical and interdisciplinary research.