Asymptotic Analysis - Volume 47, issue 1-2

Authors: Pierfelice, Vittoria

Article Type: Research Article

Abstract: We prove Strichartz estimates for the Schrödinger equation perturbed with a small time dependent potential V(t,x) belonging to the weak Lebesgue space L∞ t L(n/2,∞) x . We also consider the heat equation perturbed with a time independent singular potential V(x)∈L(n/2,∞) and we prove both the maximum principle and the Strichartz estimates under a suitable smallness condition on the negative part of the potential.

Citation: Asymptotic Analysis, vol. 47, no. 1-2, pp. 1-18, 2006

Authors: Baklouti, Hamadi | Mnif, Maher

Article Type: Research Article

Abstract: We study the resonances of semi-classical Schrödinger operator in the one-dimensional case. We consider potential with a degenerate maximum of quartic type. We first get a precise asymptotic expansion of the scattering matrix. Then we give the asymptotic expansion of resonances near the barrier maximum outside a ball of radius 𝒪(h4/3 ). Finally we describe the asymptotic expansion of resonances inside the ball. Résumé. Nous étudions les résonances de l'opérateur de Schrödinger semi-classique à une variable. Nous considérons le cas d'un potentiel présentant un maximum dégénéré de type quartique. Nous obtenons une asymptotique précise de la matrice de scattering. …Nous donnons aussi l'asymptotique des résonances proche du sommet de la barrière mais restant en dehors d'un disque de rayon 𝒪(h4/3 ). Nous décrivons enfin le développement asymptotique des résonances à l'intérieur du disque. Show more

Citation: Asymptotic Analysis, vol. 47, no. 1-2, pp. 19-48, 2006

Authors: Ramm, A.G.

Article Type: Research Article

Abstract: Let F(uε )+ε(uε −w)=0 (1) where F is a nonlinear operator in a Hilbert space H, w∈H is an element, and ε>0 is a parameter. Assume that F(y)=0, and F′(y) is not a boundedly invertible operator. Sufficient conditions are given for the existence of the solution to (1) and for the convergence lim ε→0 ‖uε −y‖=0. An example of applications is considered. In this example F is a nonlinear integral operator.

Keywords: nonlinear operator equations, bifurcation, singular perturbation, integral equations

Citation: Asymptotic Analysis, vol. 47, no. 1-2, pp. 49-54, 2006

Authors: Edward, Julian | Tebou, Louis

Article Type: Research Article

Abstract: In this paper we study the null-controllability of a beam equation with hinged ends and structural damping, the damping depending on a positive parameter. We prove that this system is exactly null controllable in arbitrarily small time. This result is proven using a combination of Ingham-type inequalities, adapted for complex frequencies, and exponential decay on various frequency bands. We then let the damping parameter tend to zero and we recover an earlier null-controllability result for the undamped beam equation.

Citation: Asymptotic Analysis, vol. 47, no. 1-2, pp. 55-83, 2006

Authors: Salort, Delphine

Article Type: Research Article

Abstract: We consider in dimension 1 the Liouville equation associated to a nonnegative metric g∈𝒞2 b ($\mathbb {R}$ ). We prove global in time dispersion and Strichartz estimates with a weight dependent on g. This weighted norm appears because of a phenomenon of concentration of trajectories when g is close to 0.

Citation: Asymptotic Analysis, vol. 47, no. 1-2, pp. 85-94, 2006

Authors: Xie, Feng | Han, Maoan | Zhang, Weijiang

Article Type: Research Article

Abstract: In this paper, a class of three-dimensional singularly perturbed systems of ordinary differential equations with two fast variables are considered. Sufficient conditions for the existence of canards are obtained by asymptotic methods.

Keywords: canards, slow manifolds, singular perturbations

Citation: Asymptotic Analysis, vol. 47, no. 1-2, pp. 95-106, 2006

Authors: Georgiev, Vladimir | Tarulli, Mirko

Article Type: Research Article

Abstract: We consider some scale invariant generalizations of the smoothing estimates for the free Schrödinger equation obtained by Kenig, Ponce and Vega (Ann. Inst. H. Poincaré Anal. Non Lineaire 10(3) (1993), 255–288; Invent. Math. 134(3) (1998), 489–545). Applying these estimates and using appropriate commutator estimates, we obtain similar scale invariant smoothing estimates for perturbed Schrödinger equation with small magnetic potential.

Keywords: Schrödinger equation, smoothing estimates, magnetic potential

Citation: Asymptotic Analysis, vol. 47, no. 1-2, pp. 107-138, 2006

Authors: Gustafsson, Björn | Mossino, Jacqueline

Article Type: Research Article

Abstract: The aim of this paper is to extend, to the linear elasticity system, the asymptotic analysis by compensated compactness previously developed by the authors for the linear diffusion equation. For simplicity, we restrict ourselves to stratified media. In the case of sole homogenization we recover the classical result of W.H. Mc Connel, deriving explicitly the effective elasticity tensor for stratified media. Here we give a new proof of his result, based on compensated compactness and on a technique of decomposing matrices. As for the case of simultaneous homogenization and reduction of dimension, we perform the asymptotic analysis, as the thickness …tends to zero, of a three-dimensional laminated thin plate having an anisotropic, rapidly oscillating elasticity tensor. The limit problem is presented in three different ways, the final formulation being a fourth-order problem on the two-dimensional plate, with explicitly given elasticity tensors and effective source terms. Show more

Keywords: nonperiodic homogenization, reduction of dimension, theory of plates, compensated compactness, laminated or stratified materials

Citation: Asymptotic Analysis, vol. 47, no. 1-2, pp. 139-169, 2006