Radial stability conditions of self-gravitating, finite disks in non uniform rotations are obtained by minimizing their energy keeping, mass and circulations constant. These global stability conditions are necessary and sufficient since no  gyroscopic terms appear in the perturbed equations. This energy principle is applied to a class of finite   differentially rotating disks with adiabatic index γ>2 and with derivatives of densities that go smoothly to zero at the boundary. The global stability limits fit remarkably well Toomre's necessary conditions for local stability. I also find that for disks with adiabatic index γ=2, Toomre's necessary local criterion is indeed a sufficient global condition for stability.

Asher Yahalom, “Stability of Radial Perturbations for Non-Uniformly Rotating Self-Gravitating, Finite, Gaseous Disks” Physics Letters A 373 pages 1170-1176 (2009) http://dx.doi.org/10.1016/j.physleta.2009.01.060.