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On the partial differential equations of electrostatic MEMS devices III: refined touchdown behavior, in preparation
"... We analyze the nonlinear elliptic problem ∆u = λf(x) (1+u) 2 on a bounded domain Ω of R N with Dirichlet boundary conditions. This equation models a simple electrostatic MicroElectromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a ..."
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Cited by 46 (8 self)
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We analyze the nonlinear elliptic problem ∆u = λf(x) (1+u) 2 on a bounded domain Ω of R N with Dirichlet boundary conditions. This equation models a simple electrostatic MicroElectromechanical 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 snapthrough may occur when it exceeds a certain critical value λ ∗ (pullin voltage). This creates a socalled “pullin 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 [11] and [12]. 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 steadystate 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,
Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains
, 2010
"... We examine the regularity of the extremal solution of the nonlinear eigenvalue problem ∆ 2 u = λf(u) on a general bounded domain Ω in R N, with the Navier boundary condition u = ∆u = 0 on ∂Ω. Here λ is a positive parameter and f is a nondecreasing nonlinearity with f(0) = 1. We give general pointw ..."
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Cited by 17 (6 self)
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We examine the regularity of the extremal solution of the nonlinear eigenvalue problem ∆ 2 u = λf(u) on a general bounded domain Ω in R N, with the Navier boundary condition u = ∆u = 0 on ∂Ω. Here λ is a positive parameter and f is a nondecreasing nonlinearity with f(0) = 1. We give general pointwise bounds and energy estimates which show that for any convex and superlinear nonlinearity f, the extremal solution u ∗ is smooth provided N ≤ 5. f(t)f • If in addition lim inf t→+∞ ′ ′ (t) (f ′) 2 (t)> 0, then u ∗ is regular for N ≤ 7. • On the other hand, if γ: = lim sup t→+∞ f(t)f ′ ′ (t) (f ′ ) 2 (t) < +∞, then the same holds for N < 8 γ. It follows that u ∗ is smooth if f(t) = et and N ≤ 8, or if f(t) = (1 + t) p and N < 8p p−1. We also show that if f(t) = (1 − t) −p, p> 1 and p = 3, then u ∗ is smooth for N ≤ 8p p+1. We note that while these are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established for Dirichlet problems on radial domains, e.g., u ∗ is smooth for N ≤ 12 when f(t) = et [11], and for N ≤ 8 when f(t) = (1 − t) −2 [9] (see also [22]).
The critical dimension for a fourth order elliptic problem with singular nonlinearity
, 2008
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On the HénonLaneEmden conjecture
, 2011
"... We consider the problem of nonexistence of solutions for the following HénonLaneEmden system j −∆u = x ..."
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Cited by 9 (3 self)
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We consider the problem of nonexistence of solutions for the following HénonLaneEmden system j −∆u = x
Linear instability of entire solutions for a class of nonautonomous elliptic equations
 IN: PROCEEDINGS OF ROYAL SOCIETY EDINBURGH SECT. A
, 2008
"... We study the effect of the potential yα on the stability of entire solutions for elliptic equations on RN, N 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet no ..."
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Cited by 9 (4 self)
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We study the effect of the potential yα on the stability of entire solutions for elliptic equations on RN, N 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet nonlinear eigenvalue problem on bounded domains. As a byproduct of our results, we will improve the known results about the regularity of such solutions.
On stable entire solutions of semilinear elliptic equations with weights
 Proc. Amer. Math. Soc
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Degenerate elliptic equations with singular nonlinearities
 CALCULUS OF VARIATIONS
, 2007
"... The behavior of the “minimal branch ” is investigated for quasilinear eigenvalue problems involving the pLaplace operator, considered in a smooth bounded domain of RN, and compactness holds below a critical dimension N #. The nonlinearity f (u) lies in a very general class and the results we prese ..."
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Cited by 7 (7 self)
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The behavior of the “minimal branch ” is investigated for quasilinear eigenvalue problems involving the pLaplace operator, considered in a smooth bounded domain of RN, and compactness holds below a critical dimension N #. The nonlinearity f (u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of pLaplace operator, for p = 2 it is crucial to define a suitable notion of semistability: the functional space we introduce in the paper seems to be the natural one and yields to a spectral theory for the linearized operator. For the case p = 2, compactness is also established along unstable branches satisfying suitable spectral information. The analysis is based on a blowup argument and stronger assumptions on the nonlinearity f (u) are required.
Regularity of the extremal solution in a MEMS model with advection
, 2008
"... We consider the regularity of the extremal solution of the nonlinear eigenvalue problem j λ ..."
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Cited by 7 (4 self)
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We consider the regularity of the extremal solution of the nonlinear eigenvalue problem j λ
Global and touchdown behaviour of the generalized MEMS device equation
"... Abstract. We will prove the local and global existence of solutions of the generalized microelectromechanical system (MEMS) equation ut = ∆u + λf(x)/g(u), u < 1, in Ω × (0, ∞), u(x, t) = 0 on ∂Ω × (0, ∞), u(x,0) = u0 in Ω, where Ω ⊂ R n is a bounded domain, λ> 0 is a constant, 0 ≤ f ∈ C α (Ω ..."
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Cited by 5 (4 self)
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Abstract. We will prove the local and global existence of solutions of the generalized microelectromechanical system (MEMS) equation ut = ∆u + λf(x)/g(u), u < 1, in Ω × (0, ∞), u(x, t) = 0 on ∂Ω × (0, ∞), u(x,0) = u0 in Ω, where Ω ⊂ R n is a bounded domain, λ> 0 is a constant, 0 ≤ f ∈ C α (Ω), f ≡ 0, for some constant 0 < α < 1, 0 < g ∈ C 2 ((−∞,1)) such that g ′ (s) ≤ 0 for any s < 1 and u0 ∈ L 1 (Ω) with u0 ≤ a < 1 for some constant a. We prove that there exists a constant λ ∗ = λ ∗ (Ω, f, g)> 0 such that the associated stationary problem has a solution for any 0 ≤ λ < λ ∗ and has no solution for any λ> λ ∗. We obtain comparison theorems for the generalized MEMS equation. Under a mild assumption on the initial value we prove the convergence of global solutions to the solution of the corresponding stationary elliptic equation as t → ∞ for any 0 ≤ λ < λ ∗. We also obtain various conditions for the existence of a touchdown time T> 0 for the solution u. That is a time T> 0 such that limtրT sup Ω u(·, t) = 1. Microelectromechanical systems (MEMS) are widely used nowadays in many electronic devices including accelerometers for airbag deployment in cars, inkjet printer heads, and the device for the protection of hard disk, etc. Interested readers can read the book, Modeling MEMS and NEMS [PB], by J.A.Pelesko and D.H. Berstein for the mathematical modeling and various applications of MEMS devices. Due to the importance of MEMS devices it is important to get a detail analysis of the mathematical models of MEMS devices. In recent years there is a lot of study on the evolution and stationary equations