  We study a Newtonian model which allows us to describe some extremely flat
objects in galactic dynamics. This model is described by a partial differential
equation system called Vlasov-Poisson, whose solutions describe the temporal
evolution of a collisionless particle system in the phase space, subject to a
self interacting gravitational potential. We treat the Flat VlasovPoisson
system with an external gravitational potential induced by a fixed mass
density. The aim of this article is the study of the existence, regularity, and
stability of steady states solutions of the Flat Vlasov-Poisson system in this
case. We solved a variational problem to find minimizers for the Casimir-Energy
functional in a suitable set of functions. The minimization problem is solved
through a reduction of the original optimization problem with a scheme used in
[FR06], but instead of a concentration-compactness argument, we use a
symmetrization argument to construct a spherically symmetric solution for the
reduced problem. It was proven that this minimizer induces a solution for the
original minimization problem. The regularity of the gravitational potential
was also obtained, implying that the solutions are steady states of the Flat
Vlasov-Poisson system. The minimization problem also works as a key to give us
a similar non-linear stability result.
