Asymptotic Analysis - Volume 82, issue 1-2

Authors: Khrabustovskyi, Andrii

Article Type: Research Article

Abstract: We deal with operators in Rn of the form A=−1/b(x)Σk=1 n ∂/∂xk (a(x)∂/∂xk ), where a(x),b(x) are positive, bounded and periodic functions. We denote by Lper the set of such operators. The main result of this work is as follows: for an arbitrary L>0 and for arbitrary pairwise disjoint intervals (αj ,βj )⊂[0,L], j=1,…,m (m∈N) we construct the family of operators {Aε ∈Lper }ε such that the spectrum of Aε has exactly m gaps in [0,L] when ε is small enough, and these gaps tend to the intervals (αj ,βj ) as ε→0. The …idea how to construct the family {Aε }ε is based on methods of the homogenization theory. Show more

Keywords: periodic elliptic operators, spectrum, gaps, homogenization

DOI: 10.3233/ASY-2012-1131

Citation: Asymptotic Analysis, vol. 82, no. 1-2, pp. 1-37, 2013

Authors: Lin, Y. | Wong, R.

Article Type: Research Article

Abstract: In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn (z,N) as the degree grows to infinity. Global asymptotic formulas are obtained as n→∞, when the ratio of the parameters n/N=c is a constant in the interval (0,1). Our method is based on a modified version of the Riemann–Hilbert approach first introduced by Deift and Zhou.

Keywords: global asymptotics, discrete Chebyshev polynomials, Riemann–Hilbert problems, Airy functions

DOI: 10.3233/ASY-2012-1135

Citation: Asymptotic Analysis, vol. 82, no. 1-2, pp. 39-64, 2013

Authors: Helffer, Bernard | Kordyukov, Yuri A.

Article Type: Research Article

Abstract: We consider a magnetic Schrödinger operator Hh =(−ih∇−A¯)2 with the Dirichlet boundary conditions in a domain Ω⊂R3 , where h>0 is a small parameter. We suppose that the minimal value b0 of the module |B¯| of the vector magnetic field B¯ is strictly positive, and there exists a unique minimum point of |B¯|, which is non-degenerate. The main result of the paper is upper estimates for the low-lying eigenvalues of the operator Hh in the semiclassical limit. We also prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding …periodic setting. Show more

Keywords: magnetic Schrödinger operator, eigenvalue asymptotics, magnetic wells, semiclassical limit, spectral gaps

DOI: 10.3233/ASY-2012-1136

Citation: Asymptotic Analysis, vol. 82, no. 1-2, pp. 65-89, 2013

Authors: Montenegro, Marcelo | Lorca, Sebastián

Article Type: Research Article

Abstract: The aim of this paper is study the equation −Δu=(log (u)+λup )χ{u>0} in Ω with Dirichlet boundary condition, where 0<p<(N+2)/(N−2) and p≠1. We regularize the term log (u) for u near 0 by using a function gε (u)=−log ((u2 +εu+ε)/(u+ε)) for u≥0 which tends to log (u) as ε→0 pointwisely. When the parameter λ>0 is sufficiently large, the corresponding energy functional to the perturbed equation −Δu+gε (u)=λ(u+ )p has nontrivial critical points uε in H0 1 (Ω). Letting ε→0, then uε converges to a solution of the original problem, which is nontrivial and nonnegative. For 1<p<(N+2)/(N−2) there is at least …one nontrivial solution. While for 0<p<1, there are at least two nontrivial distinct solutions. Show more

Keywords: singular problems, multiple solutions, variational methods, a priori estimates

DOI: 10.3233/ASY-2012-1138

Citation: Asymptotic Analysis, vol. 82, no. 1-2, pp. 91-107, 2013

Authors: Yong, Lu

Article Type: Research Article

Abstract: We study semilinear Maxwell–Landau–Lifshitz systems in one space dimension. For highly oscillatory and prepared initial data, we construct WKB approximate solutions over long times O(1/ε). The leading terms of the WKB solutions solve cubic Schrödinger equations. We show that the nonlinear normal form method of Joly, Métivier and Rauch [J. Diff. Eq. 166 (2000), 175–250] applies to this context. This implies that the Schrödinger approximation stays close to the exact solution of Maxwell–Landau–Lifshitz over diffractive times. In the context of Maxwell–Landau–Lifshitz, this extends the analysis of Colin and Lannes [Discrete and Continuous Dynamical Systems 11(1) (2004), 83–100] from times O(|ln ε|) …up to O(1/ε). Show more

Keywords: diffractive optics, Maxwell–Landau–Lifshitz system, Schrödinger equation

DOI: 10.3233/ASY-2012-1140

Citation: Asymptotic Analysis, vol. 82, no. 1-2, pp. 109-137, 2013

Authors: Canevari, Giacomo | Colli, Pierluigi

Article Type: Research Article

Abstract: We are concerned with a phase field system consisting of two partial differential equations in terms of the variables thermal displacement, that is basically the time integration of temperature, and phase parameter. The system is a generalization of the well-known Caginalp model for phase transitions, when including a diffusive term for the thermal displacement in the balance equation and when dealing with an arbitrary maximal monotone graph, along with a smooth anti-monotone function, in the phase equation. A Cauchy–Neumann problem has been studied for such a system in Commun. Pure Appl. Anal. 11 (2012), 1959–1982, by proving well-posedness and regularity …results, as well as convergence of the problem as the coefficient of the diffusive term for the thermal displacement tends to zero. The aim of this contribution is rather to investigate the asymptotic behavior of the problem as the coefficient in front of the Laplacian of the temperature goes to 0: this analysis is motivated by the types III and II cases in the thermomechanical theory of Green and Naghdi. Under minimal assumptions on the data of the problems, we show a convergence result. Then, with the help of uniform regularity estimates, we discuss the rate of convergence for the difference of the solutions in suitable norms. Show more

Keywords: phase field model, initial-boundary value problem, regularity of solutions, convergence, error estimates

DOI: 10.3233/ASY-2012-1142

Citation: Asymptotic Analysis, vol. 82, no. 1-2, pp. 139-162, 2013

Authors: Merabet, I. | Nicaise, S. | Chacha, D.A.

Article Type: Research Article

Abstract: In this paper we study the asymptotic behavior of two-dimensional transmission problems for the linear Koiter model of an elastic multi-structure composed of two thin shells with the same thickness ε<<1. Contrary to the case of membrane shells, the formal limit problem, i.e., for ε=0, fails to give a convenient approximate solution, as the limit problem is ill posed. An appropriate dilation leads to an equivalent problem, for which we prove its strong convergence towards some finite limit.

Keywords: thin shells, Koiter's model, rigid junction, elastic junction, boundary layers, sensitivity

DOI: 10.3233/ASY-121144

Citation: Asymptotic Analysis, vol. 82, no. 1-2, pp. 163-185, 2013