微机电系统中的偏微分问题

This program is devoted to study some elliptic differential problems involved by Microelectromechanical Systems (MEMS). The first one is to consider MEMS problems with disturbance term, the second one is to investigate MEMS problems with gradient term, the third one is to study MEMS problems involving the fractional Laplacian, the last one is to study the fractional MEMS problems with gradient term. We will consider the pull-in voltage, the existence and stability of the minimal solution and the weak solution when the pull-in voltage is addressed. Through the study of the above MEMS problems, one can raise the issue of MEMS research methods and techniques, rich MEMS theory, on the other hand it can provide a theoretical basis for improving the MEMS devices. Finally, by studying the MEMS problems involving the fractional laplacian, we want to make a better understanding about the commonness and difference between the nonlocal elliptic problems and the second order elliptic problems.

本项目拟讨论由微机电系统（英文名：Microelectromechanical Systems，缩写为MEMS）导出的几类偏微分问题。第一类是带有扰动项的MEMS问题，第二类是带梯度项的MEMS问题，第三类是带有分数阶拉普拉斯算子的MEMS问题，第四类是带梯度项的分数阶MEMS问题。关于以上四类问题，我们拟研究其拉闸电压和极小解的存在性、稳定性，以及当给MEMS装置施加的电压为拉闸电压时，MEMS问题的弱解。通过对以上MEMS问题的研究，一方面可以提高研究MEMS问题的方法和技巧，丰富MEMS理论，另一方面可以为改进MEMS装置提供理论依据，最后对分数阶MEMS问题的研究，有利于我们进一步揭示非局部椭圆问题与二阶椭圆问题之间的共性和区别。

我们主要研究了MEMS方程，Choquard方程，Hartree方程，以及Lane-Emden方程。具体来说，我们考虑了引入二阶椭圆算子，非局部算子，以及混合型算子时，这些方程解的存在性，对称性和渐近性。对这些问题的研究，有利于我们进一步揭示非局部椭圆问题与二阶椭圆问题之间的共性和区别，丰富非线性分析和椭圆方程的基本理论。