TF}q^{-\frac2d}\right)\sqrt{\rho(x)}.$$ Our goal in this section is to prove a similar bound without the assumption that $\rho{\leqslant}1$. Coherent states [@Lieb-81b] can usually give good bounds on the kinetic energy but they do not preserve the density. The main difficulty here is to construct a state having the exact given density $\rho$. The next result says that the semi-classical approximation to the kinetic energy is an upper bound to the exact $T(\rho)$, up to some gradient corrections.

\[thm:upper\_bound\_T\] There are two constants $\kappa_1,\kappa_2>0$ depending only on the space dimension $d$ such that $$T(\rho){\leqslant}q^{-\frac2d}c_{\rm TF}\left(1+\kappa_1{\varepsilon}\right)\int_{{\mathbb{R}}^d}\rho(x)^{1+\frac2d}dx
 +\frac{\kappa_2(1+\sqrt{\varepsilon})^2}{{\varepsilon}}\int_{{\mathbb{R}}^d}|\nabla\sqrt{\rho}(x)|^2\,dx,
 \label{eq:upper_bound_T_kappa}$$ for every $\rho\in L^1({\mathbb{R}}^d,{\mathbb{R}}_+)$ with $\sqrt{\rho}\in H^1({\mathbb{R}}^d)$ and every ${\varepsilon}>0$, where $c_{\rm TF}$ is the Thomas-Fermi constant .

Note that the gradient correction in  has a better behavior in ${\varepsilon}$ than in Nam’s lower bound .

In dimension $d=1$, March and Young have given the proof of a better estimate without the parameter ${\varepsilon}$, in [@MarYou-58 Eq. (9)]: $$T(\rho){\leqslant}q^{-2}c_{\rm TF}\int_{\mathbb{R}}\rho(x)^3\,dx+\int_{\mathbb{R}}\left|\big(\sqrt{\rho}\big)'(x)\right|^2\,dx.$$ In the same paper they also state a result in 3D (for a constant $c>c_{\rm TF}$) but the proof has a mistake. This was mentioned as a conjecture in [@Lieb-83b Sec. 5.B]. Our result  can therefore be seen as a solution to the March-Young problem. 