Salvato in:
| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2304.12850 |
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Sommario:
- 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 $\sum\limits_{y\in\mathbb{Z}^{3}} φ^2(y)=m$ 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.