In 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.
