Boundary layer analysis in the semiclassical limit of a quantum drift–diffusion model

We study a singularly perturbed elliptic second order system in one space variable as it appears in a stationary quantum drift–diffusion model of a semiconductor. We prove the existence of solutions and their uniqueness as minimizers of a certain functional and determine rigorously the principal part of an asymptotic expansion of a boundary layer of those solutions. We prove analytical estimates of the remainder terms of this asymptotic expansion, and confirm by means of numerical simulations that these remainder estimates are sharp.