Isotropic steady states in galactic dynamics revised

The present paper completes our earlier results on nonlinear stability of stationary solutions of the Vlasov-Poisson system in the stellar dynamics case. By minimizing the energy under a mass-Casimir constraint we construct a large class of isotropic, spherically symmetric steady states and prove their nonlinear stability against general, i. e., not necessarily symmetric perturbations. The class is optimal in a certain sense, in particular, it includes all polytropes of finite mass with decreasing dependence on the particle energy. 1 This work was published as Guo, Y., Rein, G.: Isotropic steady states in galactic dynamics. Commun. Math. Phys. 219, 607–629 (2001). We correct an error in the analysis of the limiting “Plummer case” which was pointed out to us by Y.-H. Wan: In the proof of the former Lemma 7 we used the “identity” ∫ f∗(x,v)dv=( ∫ f(x,v)dv)∗ where ∗ denotes the symmetric decreasing rearrangement with respect to x. This is clearly false, and we modify Section 6 accordingly and also a detail in the proof of Theorem 3.