  This paper provides a variational treatment of the effect of external charges
on the free charges in an infinite free-standing graphene sheet within the
Thomas-Fermi theory. We establish existence, uniqueness and regularity of the
energy minimizers corresponding to the free charge densities that screen the
effect of an external electrostatic potential at the neutrality point. For the
potential due to one or several off-layer point charges, we also prove
positivity and a precise universal asymptotic decay rate for the screening
charge density, as well as an exact charge cancellation by the graphene sheet.
We also treat a simpler case of the non-zero background charge density and
establish similar results in that case.
