Thomas-Fermi type variational problems with low regularity

对大量相互作用的个体的集体行为进行建模是科学界最重大的挑战之一。具有非局域相互作用的非线性方程最近开始成为研究多体系统集体行为的一种新的数学范式。自作用多体系统的基本物理模型之一是密度泛函理论的Thomas-Fermi模型。我的项目将集中于托马斯-费米型模型的数学分析，这些模型具有非常奇异的非线性，导致极小值具有非常低的正则性，在某些情况下会导致不连续的极小值。我将利用变分的方法和非线性偏微分方程组理论来建立极小值的存在性，并研究它们的性质，如自由边界的正则性和极小值的自由边界正则性。这个项目是纯数学的，但与物理和生物等应用科学有很强的潜在联系。

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.