Qualitative Studies of Some Partial Differential Equations and Systems

在本项目中，主要研究者（PI）将研究在材料科学、相变和其他应用科学中出现的一些偏微分方程。 特别是， PI计划研究定常Allen-Cahn方程及其相关方程的全局（空间）性质。我们的目标是对二维空间的所有整体解进行完全分类，并对高维空间的某些整体解进行分类。 PI还将研究在多相分离分析中出现的向量值Allen-Cahn方程的极小值的奇异集。 我们的目标是严格证明一般凸域的三结点的形成，并获得三结点的动力学。 本项目还将从理论上研究一种新的水平集图像分割方法，该方法避免了数值计算中的重新初始化，有助于揭示不同材料和相界面附近的精细结构，并有助于开发更有效的图像分割和配准技术，这在医学等领域具有重要意义。

Qualitative Studies of some Partial Differential Equations and Systems.Changfeng GuiUniversity of Connecticut, StorrsAbstractIn this project the Principal Investigator (PI) will study some partial differential equations arising in material sciences, phase transition and other applied sciences. In particular, the PI plans to investigate the global (in space) properties of stationary Allen-Cahn equation and related equations. The goal is to completely classify all entire solutions for two dimensional spaces, and classify certain entire solutions in higher dimensions. The PI will also study the singular sets of minimizers of a vector valued Allen-Cahn equation which appears in the analysis of multiple phase segregation. The goal is to rigorously prove the formation of triple junction for general convex domains and to obtain the dynamics of the triple junctions. The PI will also investigate theoretically a new level set method for image segmentation which gets rid of re-initialization in numerical computations.The research will shed a light on the finer structure near interfaces between different materials and phases,and help to develop more efficient techniques for image segmentations and registrations, which are very important for medical science and other areas.