On the partial differential equations of electrostatic MEMS devices III: Refined touchdown behavior

AbstractThis paper is a continuation of [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal. 38 (2007) 1423–1449] and [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA Nonlinear Differential Equations Appl. (2008), in press], where we analyzed nonlinear parabolic problem ut=Δu−λf(x)(1+u)2 on a bounded domain Ω of RN with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at −1. Here u is modeled to describe dynamic deflection of the elastic membrane. When a voltage—represented here by λ—is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) must occur when it exceeds a certain critical value λ∗ (pull-in voltage), creating a so-called “pull-in instability” which greatly affects the design of many devices. In an effort to achieve better MEMS design, the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f(x). In this work, some a priori estimates of touchdown behavior are established, based on which the refined touchdown profiles are obtained by adapting self-similar method and center manifold analysis. Applying various analytical and numerical techniques, some properties of touchdown set—such as compactness, location and shape—are also discussed for different classes of varying permittivity profiles

Global solutions of singular parabolic equations arising from electrostatic MEMS

AbstractWe study dynamic solutions of the singular parabolic problem(P){ut−Δu=λ∗|x|α(1−u)2,(x,t)∈B×(0,∞),u(x,0)=u0(x)⩾0,x∈B,u(x,t)=0,x∈∂B, where α⩾0 and λ∗>0 are two parameters, and B is the unit ball {x∈RN:|x|⩽1} with N⩾2. Our interest is focussed on (P) with λ∗:=(2+α)(3N+α−4)9, for which (P) admits a singular stationary solution in the form S(x)=1−|x|2+α3. This equation models dynamic deflection of a simple electrostatic Micro-Electro-Mechanical-System (MEMS) device. Under the assumption u0≨︀S(x), we address the existence, uniqueness, regularity, stability or instability, and asymptotic behavior of weak solutions for (P). Given α∗∗:=4−6N+36(N−2)4, in particular we show that if either N⩾8 and α>α∗∗ or 2⩽N⩽7, then the minimal compact stationary solution uλ∗ of (P) is stable and while S(x) is unstable. However, for N⩾8 and 0⩽α⩽α∗∗, (P) has no compact minimal solution, and S(x) is an attractor from below not from above. Furthermore, the refined asymptotic behavior of global solutions for (P) is also discussed for the case where N⩾8 and 0⩽α⩽α∗∗, which is given by a certain matching of different asymptotic developments in the large outer region closer to the boundary and the thin inner region near the singularity

The Nonexistence of Vortices for Rotating Bose-Einstein Condensates in Non-Radially Symmetric Traps

We consider ground states of rotating Bose-Einstein condensates with attractive interactions in non-radially harmonic traps $V(x)=x_1^2+\Lambda ^2x_2^2$, where $0<\Lambda \not =1$ and $x=(x_1, x_2)\in R^2$. For any fixed rotational velocity $0\le \Omega <\Omega ^*:=2\min \{1, \Lambda\}$, it is known that ground states exist if and only if $a<a^*$ for some critical constant $00$ denotes the product for the number of particles times the absolute value of the scattering length. We analyze the asymptotic expansions of ground states as $a\nearrow a^*$, which display the visible effect of $\Omega$ on ground states. As a byproduct, we further prove that ground states do not have any vortex in the region $R(a):=\{x\in R^2:\,|x|\le C (a^*-a)^{-\frac{1}{12}}\}$ as $a\nearrow a^*$ for some constant $C>0$, which is independent of $0<a<a^*$.Comment: 29 pages, this is the revised versio