Qualitative Studies of Some Partial Differential Equations and Systems

一些偏微分方程和系统的定性研究。在本项目中，首席研究员（PI）将研究材料科学、相变和其他应用科学中出现的偏微分方程。特别是，PI计划研究平稳Allen-Cahn方程和相关方程的全局（空间）性质。目标是对二维空间的所有完整解进行完全分类，并对高维空间的某些完整解进行分类。PI还将研究在多相偏析分析中出现的矢量值Allen-Cahn方程的奇异极小集。目的是严格证明一般凸域的三重结的形成，并获得三重结的动力学性质。PI还将从理论上研究一种新的水平集图像分割方法，该方法可以消除数值计算中的重新初始化。该研究将揭示不同材料和相之间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，这对医学和其他领域具有重要意义。

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.