  The purpose of this paper is to mathematically investigate the formation of a
plasma sheath, and to analyze the Bohm criterions which are required for the
formation. Bohm derived originally the (hydrodynamic) Bohm criterion from the
Euler-Poisson system. Boyd and Thompson proposed the (kinetic) Bohm criterion
from a kinetic point of view, and then Riemann derived it from the
Vlasov-Poisson system. In this paper, we prove the solvability of boundary
value problems of the Vlasov-Poisson system. On the process, we see that the
kinetic Bohm criterion is a necessary condition for the solvability. The
argument gives a simpler derivation of the criterion. Furthermore, the
hydrodynamic criterion can be derived from the kinetic criterion. It is of
great interest to find the relation between the solutions of the Vlasov-Poisson
and Euler-Poisson systems. To clarify the relation, we also study the delta
mass limit of solutions of the Vlasov-poisson system.
