A new variational approach to the stability of gravitational systems
  We consider the three dimensional gravitational Vlasov Poisson system which
describes the mechanical state of a stellar system subject to its own gravity.
A well-known conjecture in astrophysics is that the steady state solutions
which are nonincreasing functions of their microscopic energy are nonlinearly
stable by the flow. This was proved at the linear level by several authors
based on the pioneering work by Antonov in 1961. Since then, standard
variational techniques based on concentration compactness methods as introduced
by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of
stationary solutions of ground state type.
  In this paper, inspired by pioneering works from the physics litterature
(Lynden-Bell 94, Wiechen-Ziegler-Schindler MNRAS 88, Aly MNRAS 89), we use the
monotonicity of the Hamiltonian under generalized symmetric rearrangement
transformations to prove that non increasing steady solutions are local
minimizer of the Hamiltonian under equimeasurable constraints, and extract
compactness from suitable minimizing sequences. This implies the nonlinear
stability of nonincreasing anisotropic steady states under radially symmetric
perturbations.
