  We describe the structure of solutions of the kinetic Fokker-Planck equations
in domains with boundaries near the singular set in one-space dimension. We
study in particular the behaviour of the solutions of this equation for
inelastic boundary conditions which are characterized by means of a coefficient
$r$ describing the amount of energy lost in the collisions of the particles
with the boundaries of the domain. A peculiar feature of this problem is the
onset of a critical exponent rc which follows from the analysis of McKean (cf.
[26]) of the properties of the stochastic process associated to the
Fokker-Planck equation under consideration. In this paper, we prove rigorously
that the solutions of the considered problem are nonunique if $r < r_c$ and
unique if $r_{c}<r\leq 1$. In particular, this nonuniqueness explains the
different behaviours found in the physics literature for numerical simulations
of the stochastic differential equation associated to the Fokker-Planck
equation. In the proof of the results of this paper we use several asymptotic
formulas and computations in the companion paper [16].
