Existence and semiclassical limit for the quantum drift-diffusion model

In this paper, we construct a weak solution to the unipolar quantum drift-diffusion model in a bounded domain with Neumann boundary conditions, thereby extending an existence result in Gianazza et al. (Arch Rational Mech Anal 194:133–220, 2009) to the case where the boundary of the domain is only Lipschitz. We also obtain the limit of the solution as the scaled Planck constant (varepsilon ) in the problem goes to (0) .