Homogenization of a class of nonlinear eigenvalue problems

2006 ◽  
Vol 136 (1) ◽  
pp. 7-22 ◽  
Author(s):  
L. Baffico ◽  
C. Conca ◽  
M. Rajesh
Keyword(s):  
Asymptotic Behaviour ◽  
Limit Point ◽  
Eigenvalue Problems ◽  
Elliptic Operators ◽  
First Eigenvalue ◽  
Limit Operator ◽  
Nonlinear Eigenvalue Problems ◽  
Convex Functionals ◽  
Homogeneous Convex ◽  
Nonlinear Eigenvalue

In this article we study the asymptotic behaviour of the eigenvalues of a family of nonlinear monotone elliptic operators of the form Aε = −div(aε (x, ∇u)), which are sub-differentials of even, positively homogeneous convex functionals, under the assumption that the operators G-converge to an operator Ahom = −div(ahom(x, ∇u)). We show that any limit point λ of a sequence of eigenvalues λε is an eigenvalue of the limit operator Ahom, where λε is an eigenvalue corresponding to the operator Aε. We also show the convergence of the sequence of first eigenvalues to the corresponding first eigenvalue of the homogenized operator.

Asymptotics of some nonlinear eigenvalue problems modelling a MEMS Capacitor. Part II: multiple solutions and singular asymptotics

2010 ◽  
Vol 22 (2) ◽  
pp. 83-123 ◽  
Author(s):  
A. E. LINDSAY ◽  
M. J. WARD
Keyword(s):  
Singular Perturbation ◽  
Asymptotic Behaviour ◽  
Unit Ball ◽  
Unit Disk ◽  
Eigenvalue Problems ◽  
Maximal Solution ◽  
Solution Branch ◽  
Nonlinear Eigenvalue Problems ◽  
Membrane Problem ◽  
Nonlinear Eigenvalue

Some nonlinear eigenvalue problems related to the modelling of the steady-state deflection of an elastic membrane associated with a Micro-Electromechanical System capacitor under a constant applied voltage are analysed using formal asymptotic methods. These problems consist of certain singular perturbations of the basic membrane nonlinear eigenvalue problem Δu = λ/(1 + u)2 in Ω with u = 0 on ∂Ω, where Ω is the unit ball in 2. It is well known that the radially symmetric solution branch to this basic membrane problem has an infinite fold-point structure with λ → 4/9 as ϵ ≡ 1 − ||u||∞ → 0+. One focus of this paper is to develop a novel singular perturbation method to analytically determine the limiting asymptotic behaviour of this infinite fold-point structure in terms of two constants that must be computed numerically. This theory is then extended to certain generalisations of the basic membrane problem in the N-dimensional unit ball. The second main focus of this paper is to analyse the effect of two distinct perturbations of the basic membrane problem in the unit disk resulting from either a bending energy term of the form −δΔ2u to the operator, or inserting a concentric inner undeflected disk of radius δ. For each of these perturbed problems, it is numerically shown that the infinite fold-point structure for the basic membrane problem is destroyed when δ > 0, and that there is a maximal solution branch for which λ → 0 as ϵ ≡ 1 − ||u||∞ → 0+. For δ > 0, a novel singular perturbation analysis is used in the limit ϵ → 0+ to construct the limiting asymptotic behaviour of the maximal solution branch for the biharmonic problem in the unit slab and the unit disk, and for the annulus problem in the unit disk. The asymptotic results for the bifurcation curves are shown to compare very favourably with full numerical results.

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

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