On the \(N\)-dimensional stationary drift-diffusion semiconductor equations
type: Item
instance of: scholarly article
title: On the \(N\)-dimensional stationary drift-diffusion semiconductor equations
zbMATH Open document ID: 0983.35137
DOI: 10.1016/S0362-546X(99)00184-4
author: Jishan Fan
author: Hongwei Wu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-04-09T00:00:00Z
review text: The paper under review is concerned with the following system of PDE's in a bounded domain \(\Omega \subset \mathbb{R}^{N}\): \[ \begin{aligned} -&\operatorname {div} (\sigma (u)\nabla \psi)= p-n+f ,\tag{1}\\ -&\operatorname {div} J_{n}= r(n,p)(1-np)+g, \tag{2}\\ &\operatorname {div} J_{p}= r(n,p)(1-np)+g, \tag{3}\\ -&\operatorname {div}(k(u)\nabla u)= \operatorname {div} (a(\psi)J_{n}+b(\psi)J_{p}), \tag{4} \end{aligned} \] where \[ J_{n}=D_{1}\nabla n - \mu_{1}n\sigma (u)\nabla \psi , \quad J_{p}= -D_{2}\nabla p - \mu_{2} p\sigma (u)\nabla \psi \] \parindent 0cm are the current density of the concentrations. Here the unknown functions \(\psi , n, p\), and \(T\) represent the electrostatic potential, the electron density, the hole density, and the temperature, respectively. These functions are subject to mixed boundary conditions on \(\partial \Omega\). Under physically natural conditions on the functions \(\sigma ,K, r\) as well as \(a,b\) and on the other known quantities, the authors prove the existence of at least one weak solution to (1)-(4) satisfying \[ \|\psi \|_{L^{\infty}(\Omega)}+ \|n\|_{L^{\infty}(\Omega)}+ \|p\|_{L^{\infty}(\Omega)}< + \infty . \] The method of proof consists in solving a modified system in place of (1)-(4), establishing a priori estimates on the solutions to the modified system and then carrying out the passage to limit to obtain the existence of a weak solution to the system under consideration.
reviewed by: Joachim Naumann
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 82D37
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1539296
zbMATH Keywords: semiconductor equations
zbMATH Keywords: compensated compactness
zbMATH Keywords: thermal effect
zbMATH Keywords: \(L^\infty\)-estimate
MaRDI profile type: Publication
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Wikidata QID: Q126311707
description: scientific article; zbMATH DE number 1539296