Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS
Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS
复制标题
DOI：
10.1090/cln/020
复制
发表时间：
2010-04
期刊：
影响因子：
--
通讯作者：
P. Esposito;N. Ghoussoub;Yujin Guo
中科院分区：
文献类型：
--
作者：
P. Esposito;N. Ghoussoub;Yujin Guo
微纳机电系统（MEMS和NEMS）是现代技术的重要组成部分，它将联合收割机电子学与微型机械装置相结合。这是数学模型描述'静电驱动'微机电系统，这是在这本专著。即使是简化的模型，作者处理仍然导致非常有趣的二阶和四阶非线性椭圆方程（在固定的情况下）和非线性抛物方程（在动态的情况下）。虽然非线性特征值问题-固定MEMS模型适合-是PDE的一个发展良好的领域，但这里出现的平方反比非线性类型有助于揭示奇异超临界问题及其特定挑战。除了实际的考虑，该模型是一个有趣的数学现象的丰富来源。数值方法、形式渐近分析和常微分方程方法给出了大量的信息，并指出了许多问题。然而，即使在静电MEMS的最简单的理想化版本中，人们也基本上需要现代PDE技术的全部可用武器库来进行所需的严格数学分析，这是本卷的主要目标。因此，这本专著可以作为一个先进的研究生文本的动机介绍了许多最近的方法，非线性分析和偏微分方程通过分析一组方程，具有巨大的实际意义。
Micro- and nanoelectromechanical systems (MEMS and NEMS), which combine electronics with miniature-size mechanical devices, are essential components of modern technology. It is the mathematical model describing 'electrostatically actuated' MEMS that is addressed in this monograph. Even the simplified models that the authors deal with still lead to very interesting second- and fourth-order nonlinear elliptic equations (in the stationary case) and to nonlinear parabolic equations (in the dynamic case). While nonlinear eigenvalue problems - where the stationary MEMS models fit - are a well-developed field of PDEs, the type of inverse square nonlinearity that appears here helps shed a new light on the class of singular supercritical problems and their specific challenges. Besides the practical considerations, the model is a rich source of interesting mathematical phenomena. Numerics, formal asymptotic analysis, and ODE methods give lots of information and point to many conjectures. However, even in the simplest idealized versions of electrostatic MEMS, one essentially needs the full available arsenal of modern PDE techniques to do the required rigorous mathematical analysis, which is the main objective of this volume. This monograph could therefore be used as an advanced graduate text for a motivational introduction to many recent methods of nonlinear analysis and PDEs through the analysis of a set of equations that have enormous practical significance.