weaker condition that $\Omega_N$ has an $\eta$–regular boundary, which means that $|\partial\Omega_N+B_r|{\leqslant}C|\Omega_N|\eta\left(r|\Omega_N|^{-1/3}\right)$ for all $r{\leqslant}|\Omega_N|^{1/3}/C$, with $\eta(t)\to0$ when $t\to0^+$. The condition $\delta_N|\Omega_N|^{1/3}\to{\infty}$ is then replaced by $\delta_N^{-2}\eta(\delta_N|\Omega_N|^{-1/3})\to0$.

The Local Density Approximation
-------------------------------

We are now able to state our main result.

\[thm:LDA\] Let $p>3$ and $0<\theta<1$ such that $$2{\leqslant}p\theta{\leqslant}1+\frac{p}{2}.
 \label{eq:hyp_p_theta}$$ There exists a constant $C=C(p,\theta,q)$ such that $$\begin{gathered}
 \left|E(\rho)- \int_{{\mathbb{R}}^3}e_{\rm UEG}\big(\rho(x)\big)\,dx\right|
 {\leqslant}{\varepsilon}\int_{{\mathbb{R}}^3}\big(\rho(x)+\rho(x)^2\big)\,dx\\
 +\frac{C(1+{\varepsilon})}{{\varepsilon}}\int_{{\mathbb{R}}^3}|\nabla\sqrt\rho(x)|^2\,dx+\frac{C}{{\varepsilon}^{4p-1}}\int_{{\mathbb{R}}^3}|\nabla\rho^\theta(x)|^p\,dx
 \label{eq:LDA_main_estim}\end{gathered}$$ for every ${\varepsilon}>0$ and every non-negative density $\rho\in L^1({\mathbb{R}}^3)\cap L^2({\mathbb{R}}^3)$ such that $\nabla \sqrt{\rho}\in L^2({\mathbb{R}}^3)$ and $\nabla\rho^\theta\in L^p({\mathbb{R}}^3)$.

The constant $C=C(p,\theta,q)$ in our estimate  depends on the number of spin states $q$ ($q=2$ for electrons), in addition to the parameters $p$ and $\theta$. It diverges when $p\to 3^+$. If $p\to 3^+$ then we can take $\theta\to 5/6^-$. Our estimate therefore applies to densities $\rho$ with compact support, which vanish at the boundary of their support like $\delta(x)^a$ with $a>4/5$, where $\delta(x)={\rm d}(x,\partial\rho^{-1}(\{0\}))$. In particular, densities which vanish linearly are allowed. Our proof allows one to consider more singular densities, that is, to relax the constraint that $\theta p{\leqslant}1+p/2$, but then the power of ${\varepsilon}$ deteriorates.

Our estimate  is certainly not optimal and it is an interesting challenge to improve it. We conjecture that a similar inequality 