The National Science Foundation awarded grant #1901914 for $189,059 to Professor Changfeng Gui for support of the project “*Qualitative Study of the Mean Field Equation and Allen-Cahn Equation*“. This award starts August 15, 2019 and ends July 31, 2022.

The mean field equation and the Allen-Cahn equation are two important types of nonlinear partial differential equations (PDEs) which have arisen in the study of several physical phenomena such as Electroweak theory and Chern-Simons-Higgs quantum field theories, statistical mechanics of two-dimensional turbulence, phase separation and phase transition, etc. The mean field equation is also related to the rigidity of Hawking Mass in the study of general relativity as well as to self-gravitating strings for a massive W-boson model coupled to Einstein theory in account of gravitational effects in cosmology. An important aspect of the Allen-Cahn equation is the display of interfaces separating different physical regions of interests. Such interfaces often share a significant feature seen in soap bubbles, or minimal surfaces in mathematical terminology. The equation has also found applications in many other area of sciences and engineering such as astrophysics and image processing. These equations also provide excellent models for training students and junior researchers in interdisciplinary research involving mathematical methods for sciences and engineering. The PI proposes to engage in both research and training aspects of these equations and to involve students at both undergraduate level and graduate level in interdisciplinary research. Postdoctoral fellows and junior researchers will also participate and be trained in the project.

The PI plans to investigate the Sphere Covering Inequality (SCI) recently discovered by the PI and his collaborator, including its generalizations and applications, in particular to the mean field equation and its type. The SCI connects geometry to analysis and has become a powerful tool in the study of two dimensional problems in nonlinear PDEs. The PI intends to extend it to high dimensions. For the Allen-Cahn equation, the PI will focus on the existence of special 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. The PI intends to use various identities as well as Morse index information to develop new approach for these non monotone, non minimizing solutions. The long term goal is to understand completely entire solutions for both scalar and vector-valued Allen-Cahn equations and the stability and dynamics of triple junctions or quadruple junctions. The nodal sets or singularities of the solutions of Allen-Cahn equation will receive special attention in the study since they not only play an important role in the theoretic analysis of the equation, they also represent in applications the interfaces or junctions of interfaces of different phases or grain boundaries in materials such as crystalline alloys.

This award reflects NSF’s statutory mission and has been deemed worthy of support through evaluation using the Foundation’s intellectual merit and broader impacts review criteria.