MEMS模型及其他非线性本征值问题中拉入距离的估计
Estimates on Pull-in Distances in MEMS Models and other Nonlinear Eigenvalue Problems
摘要：
受微机电系统（MEMS）数学模型的启发，给出了$\IR^N$中光滑有界区域$ Ω$上$-Δu = λf(x) F(u)$形式的非线性本征值问题的极小解的上下$L^\infty$估计.我们主要感兴趣的{ \it拉入距离}，这是$L^\infty-$规范的极值解决方案$u^*$和它如何依赖于域的几何形状，空间的维数，和所谓的{ \it介电常数分布} $f$。特别是，我们的研究结果提供了各种观察到的现象的数学证明，以及严格的推导的Pelesko \cite{P}，郭潘沃德\cite{GPW}和其他人的MEMS非线性$F(u)=\frac{1}{(1-u)^2}$和幂律介电常数分布$f(x)=|x|^α$的情况下获得的几个估计数值。
英文摘要：
Motivated by certain mathematical models for Micro-Electro-Mechanical Systems (MEMS), we give upper and lower $L^\infty$ estimates for the minimal solutions of nonlinear eigenvalue problems of the form $-Δu = λf(x) F(u)$ on a smooth bounded domain $ Ω$ in $\IR^N$. We are mainly interested in the {\it pull-in distance}, that is the $L^\infty-$norm of the extremal solution $u^*$ and how it depends on the geometry of the domain, the dimension of the space, and the so-called {\it permittivity profile} $f$. In particular, our results provide mathematical proofs for various observed phenomena, as well as rigorous derivations for several estimates obtained numerically by Pelesko \cite{P}, Guo-Pan-Ward \cite{GPW} and others in the case of the MEMS non-linearity $F(u)=\frac{1}{(1-u)^2}$ and for power-law permittivity profiles $f(x)=|x|^α$.
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