论文标题
在半线性椭圆方程和由封闭的mems模型引起的负指数上
On semilinear elliptic equation with negative exponent arising from a closed MEMS model
论文作者
论文摘要
本文与椭圆方程有关 $-ΔU= \ frac {λ} {（a-u）^p} $在连接的，有限的$ c^2 $ domain $ω$ of $ \ mathbb {r}^n $ 在零dirichlet边界条件下，其中$λ> 0 $，$ n \ geq 1 $，$ p> 0 $和$ a：\barΩ\ to [0,1] $在边界上消失，速率$ {\ rm dist}（x，x，\partialΩ）^γ$ 以$γ> 0 $。 当$ p = 2 $和$ n = 2 $时，该方程将模拟封闭的微电动机电系统设备，弹性膜将弯曲的接地板粘贴在边界上，但在边界上进行绝缘。功能$ a $形状弯曲的接地板。 我们在本文中的目的是研究该方程最小解决方案的定性属性，当时$λ> 0 $，$ p> 0 $ 并展示$ a $的边界衰减如何在最小解决方案和引进电压上工作。特别是，我们对最小解决方案的稳定性进行了完整的分析。
This paper is concerned with the elliptic equation $-Δu=\frac{λ}{(a-u)^p}$ in a connected, bounded $C^2$ domain $Ω$ of $\mathbb{R}^N$ subject to zero Dirichlet boundary conditions, where $λ>0$, $N\geq 1$, $p>0$ and $a:\barΩ\to[0,1]$ vanishes at the boundary with the rate ${\rm dist}(x,\partialΩ)^γ$ for $γ>0$. When $p=2$ and $N=2$, this equation models the closed Micro-Electromechanical Systems devices, where the elastic membrane sticks the curved ground plate on the boundary, but insulating on the boundary. The function $a$ shapes the curved ground plate. Our aim in this paper is to study qualitative properties of minimal solutions of this equation when $λ>0$, $p>0$ and to show how the boundary decaying of $a$ works on the minimal solutions and the pull-in voltage. Particularly, we give a complete analysis for the stability of the minimal solutions.
