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中最重大的挑战之一。近年来，具有非局部相互作用的非线性方程开始成为研究多体系统集体行为的一种新的数学范式。密度泛函理论中的Thomas-Fermi模型是自相互作用多体系统的基本物理模型之一。我的项目将集中在具有非常奇异非线性的托马斯-费米型模型的数学分析上，这些模型会导致具有非常低规则性的最小化，在某些情况下会导致不连续最小化。我将利用变分法和非线性偏微分方程理论的方法来建立最小值的存在性，并研究它们的性质，如自由边界的规律性和最小值的自由边界的规律性。该项目属于纯数学领域，但与应用科学（如物理和生物学）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.