Partial differential equations of electrostatic MEMS

Micro-Electromechanical Systems (MEMS) combine electronics with micro-size mechanical devices in the process of designing various types of microscopic machinery, especially those involved in conceiving and building modern sensors. Since their initial development in the 1980s, MEMS has revolutionized numerous branches of science and industry. Indeed, MEMS-based devices are now essential components of modern designs in a variety of areas, such as in commercial systems, the biomedical industry, space exploration, telecommunications, and other fields of applications.;The subject of this thesis is the mathematical analysis combined with numerical simulations of a nonlinear parabolic problem ut = Deltau - lfx 1+u2 on a bounded domain of R N with Dirichlet boundary conditions. This equation models the dynamic deflection of a simple idealized electrostatic MEMS device, which consists of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at -1. When a voltage---represented here by lambda---is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) may occur when it exceeds a certain critical value lambda* (pull-in voltage). This creates a so-called pull-in instability which greatly affects the design of many devices. In order to achieve better MEMS design, the elastic membrane is fabricated with a spatially varying dielectric permittivity profile f(x).;The first part of this thesis is focussed on the pull-in voltage lambda* and the quantitative and qualitative description of the steady states of the equation. Applying analytical and numerical techniques, the existence of lambda* is established together with rigorous bounds. We show the existence of at least one steady state when lambda < lambda* (and when lambda = lambda* in dimension N < 8), while none is possible for lambda > lambda*. More refined properties of steady states---such as regularity, stability, uniqueness, multiplicity, energy estimates and comparison results---are shown to depend on the dimension of the ambient space and on the permittivity profile.;The second part of this thesis is devoted to the dynamic aspect of the parabolic equation. We prove that the membrane globally converges to its unique maximal negative steady-state when lambda ≤ lambda*, with a possibility of touchdown at infinite time when lambda = lambda* and N ≥ 8. On the other hand, if lambda > lambda* the membrane must touchdown at finite time T, which cannot take place at the location where the permittivity profile f(x) vanishes. Both larger pull-in distance and larger pull-in voltage can be achieved by properly tailoring the permittivity profile. We analyze and compare finite touchdown times by using both analytical and numerical techniques. When lambda > lambda*, some a priori estimates of touchdown behavior are established, based on which, we can give a refined description of touchdown profiles by adapting recently developed self-similarity methods as well as center manifold analysis. Applying various analytical and numerical methods, some properties of the touchdown set---such as compactness, location and shape---are also discussed for different classes of varying permittivity profiles f(x).;As it is often the case in science and technology, the quest for optimizing the attributes of MEMS devices according to their various uses, led to the development of mathematical models that try to capture the importance and the impact of the multitude of parameters involved in their design and production. This thesis is concerned with one of the simplest mathematical models for an idealized electrostatic MEMS, which was recently developed and popularized in a relatively recent monograph by J. Pelesko and D. Bernstein. These models turned out to be an incredibly rich source of interesting mathematical phenomena.