I am trying to recalculate '[Stability of Matter](http://www.pas.rochester.edu/~rajeev/phy246/lieb.pdf)' Paper from Lieb.


I have a problem at the last step of the proof at page 3, where Lieb calculates the minimizer. I will explain my ideas and my problem so far.


I have the following functional to be minimized on $\rho\in L^1(\mathbb R^d)$. Here $\lambda$ is a Lagrange multiplier and $\rho\geq 0$.


$h(\rho) = \frac{1}{C\_d}\left(\int\_{\mathbb R^d} dx \rho(x)^\frac{d}{d-2}\right)^\frac{d-2}{d} - Z \int\_{\mathbb R^d} dx \frac{\rho (x)}{|x|} +\lambda\left( \int\_{\mathbb R^d} dx \rho(x) -1\right)$


My idea is the following:


$\frac{\text{d}}{\text{d}t}h(\rho\_m +t \eta)|\_{t=0} =0$, where $\rho\_m$ is the minimizer and $\eta\in C\_0^\infty(\mathbb R^d)$ arbitrary.


Then I obtain the following:


$0 = \int dx \,\eta(x) \left(\frac{1}{C\_d}\|\rho\_m\|\_\frac{d}{d-2}^\frac{-2}{d-2}\rho\_m(x)^\frac{2}{d-2} - \frac{Z}{|x|} + \lambda\right)$


Then by the fundamental lemma of variations it follows that:


$\frac{1}{C\_d}\|\rho\_m\|\_\frac{d}{d-2}^\frac{-2}{d-2} \rho\_m(x)^\frac{2}{d-2} - \frac{Z}{|x|} + \lambda = 0$


Now I use that $\rho\_m\geq 0$ and therefore $\frac{Z}{|x|} - \lambda\geq 0$. Since one also sees that $\lambda$ has to be bigger or equal 0 as otherwise $\rho\_m$ wouldn't be in $L^1$. I conclude that $\rho\_m$ has the following form:


$\rho\_m (x) =\begin{cases} C\_d^\frac{d-2}{2} \|\rho\_m\|\_\frac{d}{d-2} \left(\frac{Z}{|x|} - \lambda \right)^\frac{d-2}{2} & \text{ if} |x|\leq \frac{Z}{\lambda} \\
0 & \text{ if} |x|> \frac{Z}{\lambda}
\end{cases}$


Now comes the point of my concerns: This $\rho\_m$ has compact support and therefore cannot really satisfy $0 = \int dx \,\eta(x) \left(\frac{1}{C\_d}\|\rho\_m\|\_\frac{d}{d-2}^\frac{-2}{d-2}\rho\_m(x)^\frac{2}{d-2} - \frac{Z}{|x|} + \lambda\right)$. Is there a reason why this $\rho\_m$ is however the correct minimizer?


Best wishes :)



Lieb: Stability of matter, problem with variational method
You are minimizing under the constraint that $\rho \ge 0$. Hence your variation $\rho\_m + t \eta$ might not be admissible. (Say if $\eta \ne 0$ and $\vert t\vert$ is large enough.)


The trick is to consider either any $\eta$ with $t \ge 0$, wich will give you the inequality
$$
 \frac{1}{C\_d} \Vert \rho\_m \Vert\_{d/(d - 2)}^{-2/(d - 2)} \rho\_m (x)^{2/(d - 2)} - \frac{Z}{\vert x \vert} + \lambda \ge 0.
$$
or consider $\eta \in L^{\infty} (\mathbb{R}^d) \cap L^1 (\mathbb{R}^d)$ such that 
$$
 \inf \{\rho\_m (x) : \eta (x) \ne 0\} > 0.
$$
Then, there exists $\tau > 0$ such that $u + t \eta \ge 0$ for each $t \in (-\tau, \tau)$. One concludes therefrom that if $\rho\_m (x) > 0$, then 
$$
 \frac{1}{C\_d} \Vert \rho\_m \Vert\_{d/(d - 2)}^{-2/(d - 2)} \rho\_m (x)^{2/(d - 2)} - \frac{Z}{\vert x \vert} + \lambda = 0.
$$
This brings you to the desired solution, and the function satisfies the necessary conditions of extremum even if its support is compact.


Since the regularity of $\rho\_m$ is not known a priori, it is important to take nonsmooth variations $\eta$.


