Thomas-Fermi type variational problems with low regularity

Modelling the collective behaviour of a large number of interacting individuals is one of the most significant challenges in Science. The non-linear equations with nonlocal interactions recently started to emerge as a new mathematical paradigm for the study of the collective behaviour of many-body systems. One of the fundamental physical models of self-interacting many-body system is the Thomas-Fermi model of density functional theory. My project will focus on the mathematical analysis of Thomas-Fermi type models with very singular non-linearities which lead to minimizers with very low regularity, and in some cases discontinuous minimizers. I will use methods of Calculus of Variation and non-linear PDEs theory to establish existence of minimizers, and to study their properties, such as regularity of the free boundary and free boundary regularity of the minimizers. This project is in Pure Mathematics but has strong potential connection to applied sciences, such as physics and biology.