We analyze the nonlinear elliptic problem $Δu=\frac{λf(x)}{(1+u)^2}$ on a bounded domain $Ω$ of $\R^N$ 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. When a voltage --represented here by $λ$-- is applied, the membrane deflects towards the ground plate and a snap-through may occur when it exceeds a certain critical value $λ^*$ (pull-in voltage). This creates a so-called "pull-in instability" which greatly affects the design of many devices. The mathematical model lends to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane which will be considered in forthcoming papers \cite{GG2} and \cite{GG3}. For now, we focus on the stationary equation where the challenge is to estimate $λ^*$ in terms of material properties of the membrane, which can be fabricated with a spatially varying dielectric permittivity profile $f$. Applying analytical and numerical techniques, the existence of $λ^*$ is established together with rigorous bounds. We show the existence of at least one steady-state when $λ< λ^*$ (and when $λ=λ^*$ in dimension $N< 8$) while none is possible for $λ>λ^*$. 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.
A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrow-gap model is given by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrow-gap model when the aspect ratio of the device tends to zero.
Well-posedness of a free boundary problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically conductive elastic membrane suspended above a fixed ground plate together with the electrostatic potential in the free domain between the membrane and the fixed ground plate. The electrostatic potential is harmonic in that domain and its values are held fixed along the membrane and the ground plate. The equation for the membrane deflection is a parabolic qua…
A moving boundary problem with two free boundaries modeling a two-dimensional idealized MEMS device with pull-in instability is discussed. We use a fixed point argument to show that the model possesses stationary solutions for small source voltages. We also give a rigorous evidence that solutions of the model converge towards solutions of the associated small aspect ratio equation in the vanishing aspect ratio limit.
The singular parabolic problem $u_t=Δu -\frac{λf(x)}{(1+u)^2}$ on a bounded domain $Ω$ of $R^N$ with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage --represented here by $λ$. As a byproduct, we prove that for sufficiently large $λ$, finite-time quenching must occur near the maximum point of the varying dielectric permittivity profile $f(x)$.
We consider a nonlocal parabolic model for a micro-electro-mechanical system. Specifically, for a radially symmetric problem with monotonic initial data, it is shown that the solution quenches, so that touchdown occurs in the device, in a situation where there is no steady state. It is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained. Numerical simulations illustrating the results are given.
In canonical models of Micro-Electro Mechanical Systems (MEMS), an event called touch- down whereby the electrical components of the device come into contact, is characterized by a blow up in the governing equations and a non-physical divergence of the electric field. In the present work, we derive novel regularized governing equations whose solutions remain finite at touchdown and exhibit additional dynamics beyond this initial event before eventually relaxing to new stable equilibria. We employ techniques from variational calculus, dynamical systems and singular perturbation theory to obtain…
The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system (MEMS) is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions correspondi…
An idealized electrostatically actuated microelectromechanical system (MEMS) involving an elastic plate with a heterogeneous dielectric material is considered. Starting from the electrostatic and mechanical energies, the governing evolution equations for the electrostatic potential and the plate deflection are derived from the corresponding energy balance. This leads to a free boundary transmission problem due to a jump of the dielectric permittivity across the interface separating elastic plate and free space. Reduced models retaining the influence of the heterogeneity of the elastic plate un…
We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. The model includes the harmonic electrostatic potential in the three-dimensional time-varying region between the plates along with a fourth-order semilinear parabolic equation for the elastic plate deflection which is coupled to the square of the gradient trace of the electrostatic potential on this plate. The strength of the coupling is tuned by a parameter $λ$ proportional to the square of the applied voltage. We prove that this free boundary problem is locally well…
The dynamical and stationary behaviors of a fourth-order evolution equation with clamped boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free boundary problem on a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes two positive parameters $λ$ and $\varepsilon$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criteri…
The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The m…
A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This asymptotic model inherits the dielectric properties of the insulating layer. It involves the electrostatic potential in the device and the deformation of the elastic plate defining the geometry of the device. The electrostatic potential is given by an elliptic equation with mixed boundary conditions in the possibly non-Lipschitz region between the two plate…
A free boundary problem modeling a microelectromechanical system (MEMS) consisting of a fixed ground plate and a deformable top plate is considered, the plates being held at different electrostatic potentials. It couples a second order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The validity of the model is expected to break down in finite time when the applied voltage exceeds a certain value, a finite time singularity occurring then. This result, already known for non-positive initial configurations of…
A variational approach is employed to find stationary solutions to a free boundary problem modeling an idealized electrostatically actuated MEMS device made of an elastic plate coated with a thin dielectric film and suspended above a rigid ground plate. The model couples a non-local fourth-order equation for the elastic plate deflection to the harmonic electrostatic potential in the free domain between the elastic and the ground plate. The corresponding energy is non-coercive reflecting an inherent singularity related to a possible touchdown of the elastic plate. Stationary solutions are const…
A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter $λ$ which is proportional to the square of the applied voltage difference. Existence of a stable stationary s…
We consider a free boundary problem modeling electrostatic microelectromechanical systems. The model consists of a fourth-order damped wave equation for the elastic plate displacement which is coupled to an elliptic equation for the electrostatic potential. We first review some recent results on existence and non-existence of steady-states as well as on local and global well-posedness of the dynamical problem, the main focus being on the possible touchdown behavior of the elastic plate. We then investigate the behavior of the solutions in the time singular limit when the ratio between inertial…
A parabolic free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a fourth-order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The strength of the coupling is tuned by a parameter $λ$ which is proportional to the square of the applied voltage difference. It is proven that the model is locally well-posed i…
We consider a parabolic problem with Robin boundary condition which arises when the edge of a micro-electro-mechanical-system (MEMS) device is connected with a flexible nonideal support. Then via a rigorous analysis we investigate the structure of the solution set of the corresponding steady-state problem. We show that a critical value (the pull-in voltage) exists so that the system has exactly two stationary solutions when the applied voltage is lower than this critical value, one stationary solution for applying this critical voltage, and no stationary solution above the critical voltage.
The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameter $λ$. It is shown that there is a threshold value $λ_*>0$ of the voltage parameter such that no radially symmetric stationary solution exists for $λ>λ_*$, while at least two such solutions exist for $λ\in (0,λ_*)$. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evol…
We discuss an evolution free boundary problem of mixed type with two free boundaries modeling an idealized electrostatically actuated MEMS device. While the electric potential is the solution of an elliptic equation, the dynamics of the membranes' displacement is modeled by two parabolic equations. It is shown that the model is locally well-posed in time and that solutions exist globally for small source voltages whereas non-existence holds for large voltage values. Moreover, our model possesses a steady state solution that is asymptotically stable. Finally, we show that in the vanishing aspec…
We study a nonlinear wave equation appearing as a model for a membrane (without viscous effects) under the presence of an electrostatic potential with strength $λ$. The membrane has a unique stable branch of steady states $u_λ$ for $λ\in\lbrack0,λ_{\ast}]$. We prove that the branch $u_λ$ has an infinite number of branches of periodic solutions (free vibrations) bifurcating when the parameter $λ$ is varied. Furthermore, using a functional setting, we compute numerically the branch $u_λ$ and their branches of periodic solutions. This approach is useful to validate rigorously the steady states $u…
We study the effect of the parameter $λ$, the dimension $N$, the profile $f$ and the geometry of the domain $Ω\subset\mathbb{R}^N$, on the question of uniqueness of the solutions to the following elliptic boundary value problem with a singular nonlinearity: $$ 180pt {{array}{ll} -Δu= \frac{λf(x)}{(1-u)^2} & \hbox{in}Ω0<u<1 &\hbox{in}Ωu=0 &\hbox{on}\partial Ω. {array}. 130pt (S)_{λ, f} $$ This equation has been proposed as a model for a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 below a rigid gr…
A semilinear parabolic equation with constraint modeling the dynamics of a microelectromechanical system (MEMS) is studied. In contrast to the commonly used MEMS model, the well-known pull-in phenomenon occurring above a critical potential threshold is not accompanied by a breakdown of the model, but is recovered by the saturation of the constraint for pulled-in states. It is shown that a maximal stationary solution exists and that saturation only occurs for large potential values. In addition, the existence, uniqueness, and large time behavior of solutions to the evolution equation are studie…
We study general equations modeling electrostatic MEMS devices \begin{equation} \begin{cases} \label{P} φ\big(r,- u'(r)\big)=λ\int_0^r\frac{f(s)}{g(u(s))}\,\mathrm{d}s, & r\in(0,1), \\ 0 < u(r) < 1, & r\in(0,1), \\ u(1) = 0, \tag{$P_λ$} \end{cases} \end{equation} where $φ$, $g$, $f$ are some functions on $[0,1]$ and $λ>0$ is a parameter. We obtain results on the existence and regularity of a touchdown solution to \eqref{P} and find upper and lower bounds on the respective pull-in voltage. In the particular case, when $φ(r,v) = r^α|v|^βv$, i.e., when the associated differential equation involve…
In this paper, we reformulate a mathematical model for the dynamics of an idealized electrostatically actuated MEMS device with two elastic membranes as an initial value problem for an abstract quasilinear evolution equation. Applying the Contraction Mapping Theorem, it is shown that the model is locally well-posed in time for any value of the source voltage of the device. In addition it is proven that the MEMS model considered here possesses global solutions for small source voltages whereas for large source voltages solutions of the model have a finite maximal existence time. Furthermore, we…
We prove the local and global existence of solutions of the generalized micro-electromechanical system (MEMS) equation $u_t =Δu+λf(x)/g(u)$, $u<1$, in $Ω\times (0,\infty)$, $u(x,t)=0$ on $\partialΩ\times (0,\infty)$, $u(x,0)=u_0$ in $Ω$, where $Ω\subset\Bbb{R}^n$ is a bounded domain, $λ>0$ is a constant, $0\le f\in C^α(\overlineΩ)$, $f\not\equiv 0$, for some constant $0<α<1$, $0<g\in C^2((-\infty,1))$ such that $g'(s)\le 0$ for any $s<1$ and $u_0\in L^1(Ω)$ with $u_0\le a<1$ for some constant $a$. We prove that there exists a constant $λ^{\ast}=λ^{\ast}(Ω, f,g)>0$ such that the associated stat…
We study the global solution curves, and prove the existence of infinitely many positive solutions for three classes of self-similar equations, with $p$-Laplace operator. In case $p=2$, these are well-known problems involving the Gelfand equation, the equation modeling electrostatic micro-electromechanical systems (MEMS), and a polynomial nonlinearity. We extend the classical results of D.D. Joseph and T.S. Lundgren \cite{JL} to the case $p \ne 2$, and we generalize the main result of Z. Guo and J. Wei \cite{GW} on the equation modeling MEMS.
A two-dimensional free boundary transmission problem arising in the modeling of an electrostatically actuated plate is considered and a representation formula for the derivative of the associated electrostatic energy with respect to the deflection of the plate is derived. The latter paves the way for the construction of energy minimizers and also provides the Euler-Lagrange equation satisfied by these minimizers. A by-product is the monotonicity of the electrostatic energy with respect to the deflection.
In this work we numerically compute the bifurcation curve of stationary solutions for the free boundary problem for MEMS in one space dimension. It has a single turning point, as in the case of the small aspect ratio limit. We also find a threshold for the existence of global-in-time solutions of the evolution equation given by either a heat or a damped wave equation. This threshold is what we term the dynamical pull-in value: it separates the stable operation regime from the touchdown regime. The numerical calculations show that the dynamical threshold values for the heat equation coincide wi…
We obtain the inequality $$\int_Ω|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_Ω \left( \sqrt{ |Δu(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $Ω\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{loc}(Ω)$ is postive and obeys some additional assumptions, $Δu$ is the Laplace operator, ${\cal T}_{h,C}(\cdot )$ is certain transformation of the continuous function $h(\cdot)$. We also explain how to apply such inequality to deduce regularity for solutions of nonlinear eigenvalue problems of elliptic type for degenerated PDEs, with the illustration within the model of electros…
We prove that the stationary magnetic potential vector and the electrostatic potential entering the dynamic magnetic Schrödinger equation can be Lipschitz stably retrieved through finitely many local boundary measurements of the solution. The proof is by means of a specific global Carleman estimate for the Schrödinger equation, established in the first part of the paper.
We consider positive solutions of a semilinear Dirichlet problem \[ Δu+λf(u)=0, \;\; \mbox{for $|x|<1$}, \;\; u=0 , \;\; \mbox{when $|x|=1$} \] on a unit ball in $R^n$. For four classes of self-similar equations it is possible to parameterize the entire (global) solution curve through the solution of a single initial value problem. This allows us to derive results on the multiplicity of solutions, and on their Morse indices. In particular, we easily recover the classical results of D.D. Joseph and T.S. Lundgren [6] on the Gelfand problem. Surprisingly, the situation turns out to be different f…
We perform the periodic homogenization (i.e. $\eps\to 0$) of the non-stationary Stokes-Nernst-Planck-Poisson system using two-scale convergence, where $\eps$ is a suitable scale parameter. The objective is to investigate the influence of \textsl{different boundary conditions and variable choices of scalings in $\eps$} of the microscopic system of partial differential equations on the structure of the (upscaled) limit model equations. Due to the specific nonlinear coupling of the underlying equations, special attention has to be paid when passing to the limit in the electrostatic drift term. As…
Uniqueness results are established for time-independent finite-energy electromagnetic fields which solve the nonlinear Maxwell--Born--Infeld equations in boundary-free space under the condition that either the charge or current density vanishes. In addition, it is also shown that the simpler Maxwell--Born equations admit at most a unique stationary finite-energy electromagnetic field solution, without the above condition. In these theories of electromagnetism, the following physical consequences emerge: source-free field solitons moving at speeds less than the vacuum speed of light $c$ do not …
Simulations of Kinetic Electrostatic Electron Nonlinear (KEEN) Waves with Variable Velocity Resolution Grids and High-Order Time-Splitting
Polymath Research Inc.（Polymath研究公司）；Universitat Jaume I（海梅一世大学）；Inria-Rennes (IPSO team) and IRMAR, Rennes, France（法国雷恩Inria（IPSO团队）和IRMAR）；Max-Planck-Institut für Plasmaphysik（马克斯·普朗克等离子体物理研究所）；Université de Strasbourg（斯特拉斯堡大学）；TU Munich（慕尼黑工业大学）
