  Density functional theory together with the Kohn-Sham scheme represent an
efficient framework to recover the ground state density and energy of a
many-body quantum system from an auxiliary ``non-interacting'' system (one-body
with a local potential). However, theoretical questions remain open. An
important one is related to the existence of a derivative of the Lieb
functional in some restricted (weak) sense (as pointed out by Lammert). Then, a
further one would be related to the validity of the ``non-interacting
$v$-representability'' conjecture. We gather here elements on these questions,
providing reminders and thoughts that are non-conclusive but hopefully
contribute to the reflection. This document must be considered as notes (it
does not represent an article).
