Nonlocal dynamic problems with singular nonlinearities and applications to MEMS (original) (raw)
Nonlocal Singular Problems and Applications to MEMS
—We consider fourth order nonlinear problems which describe electrostatic actuation in MicroElectroMechan-icalSystems (MEMS) both in the stationary case and in the evolution case; we prove existence, uniqueness and regularity theorems by exploiting the Near Operators Theory.
PERIODIC SOLUTIONS TO NONLOCAL MEMS EQUATIONS In loving memory of Alfredo
Combining a priori estimates with penalization techniques and an implicit function argument based on Campanato's near operators theory, we obtain the existence of periodic solutions for a fourth order integro-differential equation modelling actuators in MEMS devices. New insights are also derived for the stationary problem improving previous existence results by removing smallness assumptions on the domain.
Existence and dynamic properties of a parabolic nonlocal MEMS equation
arXiv (Cornell University), 2008
Let Ω ⊂ R n be a C 2 bounded domain and χ > 0 be a constant. We will prove the existence of constants λ N ≥ λ * N ≥ λ * (1 + χ Ω dx 1−w *) 2 for the nonlocal MEMS equation −∆v = λ/(1 − v) 2 (1 + χ Ω 1/(1 − v)dx) 2 in Ω, v = 0 on ∂Ω, such that a solution exists for any 0 ≤ λ < λ * N and no solution exists for any λ > λ N where λ * is the pull-in voltage and w * is the limit of the minimal solution of −∆v = λ/(1 − v) 2 in Ω with v = 0 on ∂Ω as λ ր λ *. Moreover λ N < ∞ if Ω is a strictly convex smooth bounded domain. We will prove the local existence and uniqueness of the parabolic nonlocal MEMS equation u t = ∆u + λ/(1 − u) 2 (1 + χ Ω 1/(1 − u) dx) 2 in Ω × (0, ∞), u = 0 on ∂Ω × (0, ∞), u(x, 0) = u 0 in Ω. We prove the existence of a unique global solution and the asymptotic behaviour of the global solution of the parabolic nonlocal MEMS equation under various boundedness conditions on λ. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when λ is large.
The existence and dynamic properties of a parabolic nonlocal MEMS equation
Nonlinear Analysis: Theory, Methods & Applications, 2011
Let Ω ⊂ R n be a C 2 bounded domain and χ > 0 be a constant. We will prove the existence of constants λ N ≥ λ * N ≥ λ * (1 + χ Ω dx 1−w *) 2 for the nonlocal MEMS equation −∆v = λ/(1 − v) 2 (1 + χ Ω 1/(1 − v)dx) 2 in Ω, v = 0 on ∂Ω, such that a solution exists for any 0 ≤ λ < λ * N and no solution exists for any λ > λ N where λ * is the pull-in voltage and w * is the limit of the minimal solution of −∆v = λ/(1 − v) 2 in Ω with v = 0 on ∂Ω as λ ր λ *. Moreover λ N < ∞ if Ω is a strictly convex smooth bounded domain. We will prove the local existence and uniqueness of the parabolic nonlocal MEMS equation u t = ∆u + λ/(1 − u) 2 (1 + χ Ω 1/(1 − u) dx) 2 in Ω × (0, ∞), u = 0 on ∂Ω × (0, ∞), u(x, 0) = u 0 in Ω. We prove the existence of a unique global solution and the asymptotic behaviour of the global solution of the parabolic nonlocal MEMS equation under various boundedness conditions on λ. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when λ is large.
Archive for Rational Mechanics and Analysis, 1991
Global weak solutions of scalar second-order quasilinear hyperbolic integrodifferential equations with singular kernels are constructed. Perturbations of rest states are shown to propagate with finite speed, smoothing effects of the solution operator are exhibited, and conditions for the asymptotic stability of rest states are given. The equations arise in viscoelasticity. O. Physical Background Consider a one-dimensional body which in its reference configuration is homogeneous and occupies the interval [0, L] Q R and which moves longitudinally. Let x-? u(x, t) be the position of the material particle x at time t; then the equations of motion are 9 " utt(x, t) = ax(X, t) + f(x, t), (0.1) where o is the density in the reference configuration, f denotes the body force, and a is the Piola-Kirchhoff stress. For notational simplicity, we assume that ,o = 1. Let us make the constitutive assumption that a has the form t a(x, t) ~ p
Nonautonomous Dynamical Systems, 2020
In this work, we present existence of mild solutions for partial integro-differential equations with state-dependent nonlocal local conditions. We assume that the linear part has a resolvent operator in the sense given by Grimmer. The existence of mild solutions is proved by means of Kuratowski’s measure of non-compactness and a generalized Darbo fixed point theorem in Fréchet space. Finally, an example is given for demonstration.
Regularity theory for fully nonlinear integro-differential equations
Communications on Pure and Applied Mathematics, 2009
We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump Lèvy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C 1,α regularity for general fully nonlinear integrodifferential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.
On Approximate Solution of Weakly-Singular Integro-dynamic Equation on Time Scales
gazi university journal of science, 2015
Many mathematical formulations of physical phenomena contain integro-dynamic equations. In this paper, we present a new and simple approach to resolve linear and nonlinear weakly-singular Volterra integro-dynamic equations of first and second order on any time scales. These equations occur in many applications shuch as in heat transfer, nuclear reactor dynamics, dynamics of linear viscoelastic materyal with long memory etc. In order to eliminate the singularity of the equation, nabla derivative is used and then transforming the given first-order integro-dynamic equations onto an firstorder dynamic equations on time scales. The validity of the method is illustrated with some examples. It has been observed that the numerical results efficiently approximate the exact solutions
An Analytic Approach to Weakly-Singular Integro-Dynamic Equation on Time Scales
2017
In this paper, we present a new and simple approach to resolve linear and nonlinear weakly-singular Volterra integro-dynamic equations of first and second order on any time scales. In order to eliminate the singularity of the equation, nabla derivative is used and then transforming the given first-order integro-dynamic equations onto an first-order dynamic equations on time scales. The validity of the method is illustrated with some examples.