{\displaystyle}\int_{{\mathbb{R}}^3}|\nabla\rho^\theta(x)|^p\,dx\ll \int_{{\mathbb{R}}^3}\big(\rho(x)+\rho(x)^2\big)\,dx.   
  \end{cases}$$ For instance for a rescaled density in the form $$\rho_N(x):=\rho\big(N^{-1/3}x\big)$$ with $\int_{{\mathbb{R}}^3}\rho=1$, we obtain after taking ${\varepsilon}=N^{-1/12}$ $$\left|F_{\rm LL}(\rho_N)- \frac{N^{\frac53}}{2}\int_{{\mathbb{R}}^3}\int_{{\mathbb{R}}^3}\frac{\rho(x)\rho(y)}{|x-y|}\,dx\,dy-N\int_{{\mathbb{R}}^3}e_{\rm UEG}\big(\rho(x)\big)\,dx\right|
 {\leqslant}C N^{\frac{11}{12}}.
 \label{eq:LDA_main_rescaled_density_intro}$$

The bound  is, to our knowledge, the first estimate of this kind on the fundamental functional $F_{\rm LL}$. Although it should be possible to extract a definite value of the constant $C$ from our proof, it is probably very large and we have not tried to do it. The factor $1/{\varepsilon}^{4p-1}$ is also quite large and it is an open problem to improve it. We hope that our work will stimulate more results on the functional $F_{\rm LL}$ in the regime of slowly varying densities.

In physics and chemistry, the *exchange-correlation energy* is defined by subtracting a kinetic energy term $T(\rho)$ from $F_{\rm LL}(\rho)$. In this paper we also derive a bound on $T(\rho)$ which, when combined with , provides a bound on the exchange-correlation energy similar to . This is explained below in Remark \[rmk:exchange\_correlation\].

In the next section we provide the precise mathematical definition of $F_{\rm LL}$ and $e_{\rm UEG}$, and we state our main theorem containing the estimate . In Section \[sec:a\_priori\] we review some known *a priori* estimates on 