变介电常数下MEMS问题的触地定位
Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity
摘要：
我们考虑了一个著名的模型的微机电系统（MEMS）的可变介电常数，基于一个抛物型方程奇异非线性。我们研究了触地或淬火现象。最近，问题是否接地可以发生在零点的介电常数的配置文件，长期以来一直保持开放，回答消极的郭和Souplet的情况下，内点，然后我们表明，接地实际上可以排除在子区域的域的介电常数是积极的，但适当的小。本文的目的是进一步研究触地定位问题，并表明，在一个空间维，人们可以得到相当定量的条件。也就是说，对于大类的典型的，一个凸块和两个凸块的介电常数的配置文件，我们发现良好的较低的估计之间的比率f和它的最大值，低于没有接地发生以外的凸块。该比率严格地获得作为合适的有限维最优化问题（具有三个或四个参数）的解，然后对其进行数值估计。相当令人惊讶的是，事实证明，该比率的值并不“小”，而是实际上高达0.3的量级，因此这可能非常适合于在实际MEMS设计中的鲁棒使用。减少到有限维最优化问题的主要工具是定量的I型，时间降落估计。后者证明了最大值原理参数，适用于一个多参数家庭的细化，非线性辅助功能与截止。
英文摘要：
We consider a well-known model for micro-electromechanical systems (MEMS) with variable dielectric permittivity, based on a parabolic equation with singular nonlinearity. We study the touchdown or quenching phenomenon. Recently, the question whether or not touchdown can occur at zero points of the permittivity profile, which had long remained open, was answered negatively by Guo and Souplet for the case of interior points, and we then showed that touchdown can actually be ruled out in subregions of the domain where the permittivity is positive but suitably small. The goal of this paper is to further investigate the touchdown localization problem and to show that, in one space dimension, one can obtain quite quantitative conditions. Namely, for large classes of typical, one-bump and two-bump permittivity profiles, we find good lower estimates of the ratio between f and its maximum, below which no touchdown occurs outside of the bumps. The ratio is rigorously obtained as the solution of a suitable finite-dimensional optimization problem (with either three or four parameters), which is then numerically estimated. Rather surprisingly, it turns out that the values of the ratio are not "small" but actually up to the order 0.3, which could hence be quite appropriate for robust use in practical MEMS design. The main tool for the reduction to the finite-dimensional optimization problem is a quantitative type I, temporal touchdown estimate. The latter is proved by maximum principle arguments, applied to a multi-parameter family of refined, nonlinear auxiliary functions with cut-off.
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