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46
Regularity of radial minimizers of reaction equations involving the pLaplacian
, 2007
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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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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
LOW DIMENSIONAL INSTABILITY FOR SEMILINEAR AND QUASILINEAR PROBLEMS IN RN
"... Abstract. Stability properties for solutions of −∆m(u) = f(u) in RN are investigated, where N ≥ 2 and m ≥ 2. The aim is to identify a critical dimension N # so that every nonconstant solution is linearly unstable whenever 2 ≤ N < N#. For positive, increasing and convex nonlinearities f(u), glob ..."
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Abstract. Stability properties for solutions of −∆m(u) = f(u) in RN are investigated, where N ≥ 2 and m ≥ 2. The aim is to identify a critical dimension N # so that every nonconstant solution is linearly unstable whenever 2 ≤ N < N#. For positive, increasing and convex nonlinearities f(u), global bounds on f f (f ′)2 allows us to find a dimension N which is optimal for exponential and power nonlinearities. In the radial setting we can deal more generally with C1−nonlinearities and the dimension N # we find is still optimal.
Linearly elastic annular and circular membranes under radial, transverse, and torsional loading. Part I: large unwrinkled axisymmetric deformations.
 Acta Mech.
, 2009
"... Abstract Three theories for determination of the equilibrium states of initially flat, linearly elastic, rotationally symmetric, taut membranes are considered: Föpplvon Kármán theory, Reissner's theory, and a new generalization of Reissner's theory that does not restrict the strains to b ..."
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Abstract Three theories for determination of the equilibrium states of initially flat, linearly elastic, rotationally symmetric, taut membranes are considered: Föpplvon Kármán theory, Reissner's theory, and a new generalization of Reissner's theory that does not restrict the strains to be small. Attention is focused on annular membranes, but circular membranes are also treated. Large deformations are allowed, and the equilibrium equations are written in terms of transverse, radial, and circumferential displacements. Problems considered include radial stretching, transverse displacement of the inner edge, an adhesive punch pulloff test on a circular blister, transverse pressure, ponding of annular and circular membranes, a vertical distributed load with a vertically sliding outer membrane edge, pullin (snapdown, jumptocontact) instability of a MEMS device, torsion of the inner or outer edge of a stretched membrane, and a combination of radial stretching, vertical displacement, and torsion. Results for the three theories are compared. Closedform solutions are available in a few cases, but usually a shooting method is utilized to obtain numerical solutions for displacements, strains, and stresses. Conditions for the onset of wrinkling are determined. In the second part of this twopart study, small vibrations about equilibrium configurations are analyzed.
On MEMS equation with fringing field
"... We consider the MEMS equation with fringing field −∆u = λ(1 + δ∇u  2)(1 − u) −2 in Ω, u = 0 on ∂Ω where λ, δ> 0 and Ω ⊂ Rn is a smooth and bounded domain. We show that when the fringing field exists (i.e. δ> 0), given any µ> 0, we have uniform upper bound of classical solutions u away fro ..."
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We consider the MEMS equation with fringing field −∆u = λ(1 + δ∇u  2)(1 − u) −2 in Ω, u = 0 on ∂Ω where λ, δ> 0 and Ω ⊂ Rn is a smooth and bounded domain. We show that when the fringing field exists (i.e. δ> 0), given any µ> 0, we have uniform upper bound of classical solutions u away from the rupture level 1 for all λ ≥ µ. Moreover, there exists λ ∗ δ> 0 such that there are at least two solutions when λ ∈ (0, λ ∗ δ); a unique solution exists when λ = λ ∗ δ; and there is no solution when λ> λ ∗ δ. This represents a dramatic change of behavior with respect to the zero fringing field case (i.e. δ = 0) and confirms the simulations in [14, 11]. Key words. MEMS, rupture, fringing field, bifurcation 2000 Mathematics Subject Classification. 35B45, 35J40
Uniqueness of solutions for an elliptic equation modeling MEMS
, 2008
"... We study the effect of the parameter λ, the dimension N, the profile f and the geometry of the domain Ω ⊂ R N, on the ..."
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We study the effect of the parameter λ, the dimension N, the profile f and the geometry of the domain Ω ⊂ R N, on the
pMEMS equation on a ball
 Methods Appl. Anal
"... Abstract. We investigate qualitative properties of the MEMS equation involving the p−Laplace operator, 1 < p ≤ 2, on a ball B in RN, N ≥ 2. We establish uniqueness results for semistable solutions and stability (in a strict sense) of minimal solutions. In particular, along the minimal branch we ..."
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Abstract. We investigate qualitative properties of the MEMS equation involving the p−Laplace operator, 1 < p ≤ 2, on a ball B in RN, N ≥ 2. We establish uniqueness results for semistable solutions and stability (in a strict sense) of minimal solutions. In particular, along the minimal branch we show monotonicity of the first eigenvalue for the corresponding linearized operator and radial symmetry of the first eigenfunction. Key words. AMS subject classifications. 35B05, 35B65, 35J70
A new model for electrostatic MEMS with two free boundaries
 J. Math. Anal. Appl
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On an ellipticparabolic MEMS model with two free boundaries. 2014, submitted; see www.arxiv.org
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Recent developments on a nonlocal problem arising in the microelectro mechanical system
 Tamkang J. Math
"... Abstract. In this paper, we study an evolution problem arises in the study of MEMS (microelectro mechanical system) device. We consider both parabolic and hyperbolic type problems. We summarize some recent results on the steady states and the global vs nonglobal existence of solutions. We also lis ..."
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Abstract. In this paper, we study an evolution problem arises in the study of MEMS (microelectro mechanical system) device. We consider both parabolic and hyperbolic type problems. We summarize some recent results on the steady states and the global vs nonglobal existence of solutions. We also list some open problems and provide a list (far from complete) of references related to this subject. 1. Description of the model: MEMS Weare interested in the dynamicdeflection of an elasticmembrane inside amicroelectro mechanical system (MEMS). Typically, the MEMS device consists of an electric membrane hanged above a rigid ground plate, connected in series with a fixed voltage source and a fixed capacitor. We refer to [29, 30] and the references therein for the physical backgrounds and its applications. In the case the distance between the plate and themembrane is relative small compared to the length of the device, the original mathematical system describing the operation of the MEMS is reduced to the following single nonlocal equation in dimensionless variables εut t +ut =∆u+ λ f (x) (1−u)2