Quasi-Neutral Limit In The Steady Euler-Poisson Equations With Insulating And Contact Boundary Conditions

The electrical neutrality of semiconductors is the basic assumption of semiconductor physics.In this paper,by studying the quasi-neutral limit problem of the one-dimensional steady EulerPoisson equation under contact boundary conditions and insulation boundary conditions,we verify mathematically the hypothesis of electrical neutrality of semiconductor materials.The main research methods and results of this paper are as follows:Firstly,based on the asymptotic expansion technique,we carry out the first-order expansion of the one-dimensional steady Euler-Poisson equation in terms of the Debye length λ in the interior of the interval and near the boundary.By using the matching technique,we obtain the boundary layer equation.Then,we prove the exponential decay of the boundary layer solution and the error estimate of the zero-order approximate solution.That is,we derive the strong convergence and the convergence rate of the asymptotic limit under the contact boundary condition.Compared with the existing results,we do not need the hypothesis of smallness of boundary conditions.Furthermore,in view of the special structure of one-dimensional steadystate Euler-Poisson equation,we consider the boundary layers at both ends and adopt a more friendly maximum principle to derive the error estimates.Secondly,we also study the quasi-neutral limit problem of the one-dimensional steady Euler-Poisson equation under the insulation boundary conditions.Employing the method of asymptotic expansion as the same as the case of contact boundary conditions,we obtain the first-order formal asymptotic expansion in terms of Debye length,and derive the boundary layer equation.Then we prove the exponential decay of the boundary layer solution.Finally,we obtain the error estimates of the first order approximate solution by using the Schauder fixed point theorem.Compared with the case of contact boundary condition,the equation has a firstorder boundary layer rather than a zero-order boundary layer in the case of insulation boundary condition.