The high field asymptotics for degenerate semiconductors: Initial and boundary layer analysis

Abstract

In a previous paper (Math. Methods Models Appl. Sci. 11 (2001), 1253–1272), the high field limit for degenerate semiconductors is analyzed by the authors. The scope of the present paper is the extention of this analysis to boundary value problems. The initial layer, modeled by a homogeneous Boltzmann equation with a frozen electric field provides an initial condition for the high field solution. The boundary layers, analyzed by means of Milne problems, are shown to provide boundary condition for the limit equation on that part of the boundary corresponding to an inflowing flux, while they connect the fluid and kinetic data on the outgoing part of the boundary. Under some monotonicity assumptions on the high field solution (which are satisfied at least for low densities), the asymptotic behaviour of the Milne problem is exhibited. These results are used to provide an error estimate for the high field asymptotics.