Qualitative Studies of Some Partial Differential Equations and Systems

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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.