Exact Results in Quantum Theory & Gravity

We discuss density functional theories of Thomas-Fermi and Thomas-Fermi-von Weizsacker type which describe the response of a single layer of graphene to an external electric charge. Mathematically, this amounts to the analysis of two nonlocal variational problems which involve Coulombic terms and a Hardy type potential. We develop the variational framework in which the proposed energy functionals admit minimizers and prove the uniqueness and regularity of the ground states for the associated Euler-Lagrange equations which involve the fractional Laplacian. In addition, we discuss positivity and decay rate of the ground states and present several open problems. This is a joint work with Jianfeng Lu (Duke) and Cyrill Muratov (NJIT, USA).