Optimal decay and regularity for a Thomas--Fermi type variational problem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929415279083520 |
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| author | Greco, Damiano |
| author_facet | Greco, Damiano |
| contents | We study existence and qualitative properties of the minimizers for a Thomas--Fermi type energy functional defined by $$E_α(ρ):=\frac{1}{q}\int_{\mathbb{R}^d}|ρ(x)|^q dx+\frac{1}{2}\iint_{\mathbb{R}^d\times\mathbb{R}^d}\frac{ρ(x)ρ(y)}{|x-y|^{d-α}}dx dy-\int_{\mathbb{R}^d}V(x)ρ(x)dx,$$ where $d\ge 2$, $α\in (0,d)$ and $V$ is a potential. Under broad assumptions on $V$ we establish existence, uniqueness and qualitative properties such as positivity, regularity and decay at infinity of the global minimizer. The decay at infinity depends in a non--trivial way on the choice of $α$ and $q$. If $α\in (0,2)$ and $q>2$ the global minimizer is proved to be positive under mild regularity assumptions on $V$, unlike in the local case $α=2$ where the global minimizer has typically compact support. We also show that if $V$ decays sufficiently fast the global minimizer is sign--changing even if $V$ is non--negative. In such regimes we establish a relation between the positive part of the global minimizer and the support of the minimizer of the energy, constrained on the non--negative functions. Our study is motivated by recent models of charge screening in graphene, where sign--changing minimizers appear in a natural way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12586 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal decay and regularity for a Thomas--Fermi type variational problem Greco, Damiano Analysis of PDEs 35J61, 35B09, 35B33, 35B40 We study existence and qualitative properties of the minimizers for a Thomas--Fermi type energy functional defined by $$E_α(ρ):=\frac{1}{q}\int_{\mathbb{R}^d}|ρ(x)|^q dx+\frac{1}{2}\iint_{\mathbb{R}^d\times\mathbb{R}^d}\frac{ρ(x)ρ(y)}{|x-y|^{d-α}}dx dy-\int_{\mathbb{R}^d}V(x)ρ(x)dx,$$ where $d\ge 2$, $α\in (0,d)$ and $V$ is a potential. Under broad assumptions on $V$ we establish existence, uniqueness and qualitative properties such as positivity, regularity and decay at infinity of the global minimizer. The decay at infinity depends in a non--trivial way on the choice of $α$ and $q$. If $α\in (0,2)$ and $q>2$ the global minimizer is proved to be positive under mild regularity assumptions on $V$, unlike in the local case $α=2$ where the global minimizer has typically compact support. We also show that if $V$ decays sufficiently fast the global minimizer is sign--changing even if $V$ is non--negative. In such regimes we establish a relation between the positive part of the global minimizer and the support of the minimizer of the energy, constrained on the non--negative functions. Our study is motivated by recent models of charge screening in graphene, where sign--changing minimizers appear in a natural way. |
| title | Optimal decay and regularity for a Thomas--Fermi type variational problem |
| topic | Analysis of PDEs 35J61, 35B09, 35B33, 35B40 |
| url | https://arxiv.org/abs/2302.12586 |