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We consider the one-dimensional Vlasov equation with an attractive cosinepotential, and its non homogeneous stationary states that are decreasingfunctions of the energy. We show that in the Sobolev space $W^{1,p}$ ($p>2$)neighborhood of such a state, all stationary states that are decreasingfunctions of the energy are stable. This is in sharp contrast with thesituation for homogeneous stationary states of a Vlasov equation, where acontrol over strictly more than one derivative is needed to ensure the absenceof unstable stationary states in a neighborhood of a reference stationary state[Z.Lin and C.Zeng, Comm.Math.Phys. {\bf 306}, 291-331 (2011)].

On the neighborhood of an inhomogeneous stable stationary solution of the Vlasov equation - Case of the Hamiltonian mean-field model