Existence and nonexistence of minimizer for Thomas-Fermi-Dirac-Von Weizsäcker model on lattice graph

Abstract

The focus of our paper is to investigate the possibility of a minimizer for the Thomas-Fermi-Dirac-von Weizsäcker model on the lattice graph ${\mathbb{Z}}^3$. The model is described by the following functional: \begin{equation*} E(φ)=\sum_{y\in\mathbb{Z}^{3}}\left(|\nablaφ(y)|^2+ (φ(y))^{\frac{10}{3}}-(φ(y))^{\frac{8}{3}}\right)+ \sum_{x,y\in\mathbb{Z}^{3}\atop ~\ y\neq x\hfill}\frac{φ^2(x)φ^2(y)}{|x-y|}, \end{equation*} with the additional constraint that $\sum _{y\in {\mathbb{Z}}^3}{\phi }^2(y)=m$ is sufficiently small. We also prove the nonexistence of a minimizer provided the mass $m$ is adequately large. Furthermore, we extend our analysis to a subset $\Omega \subset {\mathbb{Z}}^3$ and prove the nonexistence of a minimizer for the following functional: \begin{equation*} E(Ω)=|\partialΩ|+\sum_{x,y\inΩ\atop ~y\neq x\hfill}\frac{1}{|x-y|}, \end{equation*} under the constraint that $|\Omega |=V$ is sufficiently large.

21 pages