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(\varphi)=\sum_{y\in\mathbb{Z}^{3}}\left(|\nabla\varphi(y)|^2+ (\varphi(y))^{\frac{10}{3}}-(\varphi(y))^{\frac{8}{3}}\right)+ \sum_{x,y\in\mathbb{Z}^{3}\atop ~\ y\neq x\hfill}\frac{{\varphi}^2(x){\varphi}^2(y)}{|x-y|}, \end{equation*} with the additional constraint that $\sum\limits_{y\in\mathbb{Z}^{3}} {\varphi}^2(y)=m$ 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 $Ω\subset \mathbb{Z}^{3}$ 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 $|Ω|=V$ is sufficiently large.

Figures (3)

• Figure 1: The sketch of $\partial B_R$.
• Figure 2: The sketch of $\psi_n$
• Figure 3: The sketch of $B_R$ and $\Omega$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5