Vlasov-Poisson Equilibria as Solutions of a Mixed Hilbert-Stjelties Integral Inverse Problem

We present exact, weak, two species Vlasov equlibria as solutions of a mixed Stjelties-Hilbert integral inverse problem. We apply these solutions to the steady state electron rich sheath associated with a non monotonic potential profile and an asymmetric potential minimum. The electron distribution on one side of this minimum is smooth, but differs from that on the other side, which is jump discontinuous on the separatrix, but otherwise stable against the bump on tail instability: their difference is related to the distribution of the finite mass, mobile ions, which is log singular at the virtual cathode.