Linear electron stability for a bi-kinetic sheath model

We establish the linear stability of an electron equilibrium for an electrostatic and collisionless plasma in interaction with a wall. The equilibrium we focus on is called in plasma physics a Debye sheath. Specifically, we consider a two species (ions and electrons) Vlasov-Poisson-Ampère system in a bounded and one dimensional geometry. The interaction between the plasma and the wall is modeled by original boundary conditions: On the one hand, ions are absorbed by the wall while electrons are partially re-emitted. On the other hand, the electric field at the wall is induced by the accumulation of charged particles at the wall. These boundary conditions ensure the compatibility with the Maxwell-Ampère equation. A global existence, uniqueness and stability result for the linearized system is proven. The main difficulty lies in the fact that (due to the absorbing boundary conditions) the equilibrium is a discontinuous function of the particle energy, which results in a linearized system that contains a degenerate transport equation at the border.