Asymptotic Analysis - Volume 40, issue 1

Authors: Piat, V. Chiadò | Sandrakov, G.V.

Article Type: Research Article

Abstract: Methods of homogenization for variational inequalities with gradient constraints are presented. The variational inequalities are defined by a monotone operator of second order with periodic rapidly oscillating coefficients. The macroscopic homogenized (limiting) variational inequality satisfied by the limit of the solutions of the variational inequalities is deduced. Methods of variational inequalities theory and the notion of two‐scale convergence are used for passage to the limit. The relevant feature of the homogenized variational inequality is its twice‐nonlinear form, which is different from that of the initial variational inequality. A comparison with the homogenization for constrained minimization problems is given.

Citation: Asymptotic Analysis, vol. 40, no. 1, pp. 1-23, 2004

Authors: Ghisi, Marina | Gobbino, Massimo

Article Type: Research Article

Abstract: We investigate the evolution problem u"+δu'+m(|A1/2 u|2 )Au=0, u(0)=u0 ,  u'(0)=u1 , where H is a Hilbert space, A is a self‐adjoint non‐negative operator on H with domain D(A), δ>0 is a parameter, and m :[0,+∞[→[0,+∞[ is a locally Lipschitz continuous function. We prove that this problem has a unique global solution for positive times, provided that the initial data (u0 ,u1 )∈D(A)×D(A1/2 ) satisfy a suitable smallness assumption and the non‐degeneracy condition m(|A1/2 u0 |2 )>0. Moreover (u(t),u′(t),u″(t))→(u∞ ,0,0) in D(A)×D(A1/2 )×H as t→+∞, where |A1/2 u∞ |m(|A1/2 u∞ |2 )=0. These results apply to degenerate hyperbolic …PDEs with non‐local non‐linearities. Show more

Keywords: hyperbolic equations, degenerate hyperbolic equations, dissipative equations, global existence, asymptotic behaviour, Kirchhoff equations

Citation: Asymptotic Analysis, vol. 40, no. 1, pp. 25-36, 2004

Authors: Palombaro, Mariapia | Ponsiglione, Marcello

Article Type: Research Article

Abstract: We prove that for any connected open set Ω⊂$\mathbb{R}$ n and for any set of matrices K={A1 ,A2 ,A3 }⊂$\mathbb{M}$ m×n , with m≥n and rank(Ai −Aj )=n for i≠j, there is no non‐constant solution B∈L∞ (Ω,$\mathbb{M}$ m×n ), called exact solution, to the problem Div B=0 in 𝒟′(Ω,$\mathbb{R}$ m ) and B(x)∈K a.e.in Ω. In contrast, Garroni and Nesi [10] exhibited an example of set K for which the above problem admits the so‐called approximate solutions. We give further examples of this type. We also prove non‐existence of exact solutions when K is an arbitrary …set of matrices satisfying a certain algebraic condition which is weaker than simultaneous diagonalizability. Show more

Keywords: differential inclusions, phase transitions, homogenization

Citation: Asymptotic Analysis, vol. 40, no. 1, pp. 37-49, 2004

Authors: Loeper, G. | Vasseur, A.

Article Type: Research Article

Abstract: We consider in this paper a plasma subject to a strong deterministic magnetic field and we investigate the effect on this plasma of a stochastic electric field. We show that the limit behaviour, which corresponds to the transfer of energy from the electric wave to the particles (Landau phenomena), is described by a Spherical Harmonics Expansion (SHE) model.

Citation: Asymptotic Analysis, vol. 40, no. 1, pp. 51-65, 2004

Authors: Barbu, Luminiţa | Moroşanu, Gheorghe

Article Type: Research Article

Abstract: In this paper we study a model for convection–diffusion processes in which (1) the characters of both diffusion and convection change discontinuously at an internal domain point, (2) there is a small parameter ε, making it a singular perturbation problem, and (3) one of the boundary conditions is nonlinear. Specifically, the problem is (Sε ), (BCε ), (ICε ), (TCε ) formulated below. The problem is singularly perturbed with respect to the uniform convergence topology, with an internal transition layer. An asymptotic expansion of order zero for the solution is determined formally. Then some estimates for the remainder components are …established to validate our expansion. Show more

Keywords: singularly perturbed, coupled, parabolic, convection–diffusion, asymptotic expansion, internal transition layer

Citation: Asymptotic Analysis, vol. 40, no. 1, pp. 67-81, 2004

Authors: Shibata, Tetsutaro

Article Type: Research Article

Abstract: We consider the nonlinear eigenvalue problem −Δu=λf(u), u>0 in BR , u=0 on ∂BR , where BR is a ball with radius R>0 and λ>0 is a parameter. Under the appropriate conditions of f, it is known that for a given 0<ε<1, there exists (λ,u)=(λ(ε),uε ) satisfying the equation with ∫BR F(uε (x)) dx=|BR |F(u0 )(1−ε), where F(u)=∫0 u f(s) ds and u0 >0 is the smallest zero of f in R+ . Furthermore, uε (x)→u0 (x∈BR ) and λ(ε)→∞ as ε→0. This concept of parametrization of solution pair by a new parameter ε is based on the variational …structure of the equation. We establish the asymptotic formulas for λ(ε) as ε→0 with the ‘optimal’ estimate of the second term. Show more

Keywords: asymptotic formula, variational characterization of solution, simple pendulum

Citation: Asymptotic Analysis, vol. 40, no. 1, pp. 83-91, 2004