Asymptotic Analysis - Volume 62, issue 3-4

Authors: Cardoso, Fernando | Cuevas, Claudio | Vodev, Georgi

Article Type: Research Article

Abstract: We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrödinger group eit(−Δ+V) for a class of real-valued potentials V∈Ck (Rn ), V(x)=O(〈x〉−δ ), where n=4, 5, k>(n−3)/2, δ>3 if n=4 and δ>5 if n=5.

Keywords: Schrödinger equation, potential, dispersive estimates

DOI: 10.3233/ASY-2009-0916

Citation: Asymptotic Analysis, vol. 62, no. 3-4, pp. 125-145, 2009

Authors: Kim, Hyejin

Article Type: Research Article

Abstract: It is well known that solutions of classical initial-boundary problems for second-order parabolic equations depend continuously on the coefficients if the coefficients converge to their limits in a strong enough topology. In case of one spatial variable, we consider the question of the weakest possible topology providing convergence of the solutions. The problem is closely related to weak convergence of corresponding diffusion processes. Continuous Markov processes corresponding to the Feller operators Dv Du arise, in general, as limiting processes. Solutions of the parabolic equations converge, in general, to the solutions of corresponding initial-boundary problems for the limiting operator …Dv Du . Show more

Keywords: weak convergence, generalized second-order differential operators, gluing conditions, narrow tubes, discontinuous coefficients

DOI: 10.3233/ASY-2009-0918

Citation: Asymptotic Analysis, vol. 62, no. 3-4, pp. 147-162, 2009

Authors: Faraj, A. | Mantile, A. | Nier, F.

Article Type: Research Article

Abstract: We consider the stationary Schrödinger–Poisson model with a background potential describing a quantum well. The Hamiltonian of this system composes of contributions – the background potential well plus a nonlinear repulsive term – which extends on different length scales with ratio parametrized by the small parameter h. With a partition function which forces the particles to remain in the quantum well, the limit h→0 in the nonlinear system leads to different asymptotic behaviours, including spectral renormalization, depending on the dimensions 1, 2 or 3.

Keywords: Schrödinger–Poisson systems, asymptotic analysis, multiscale problems, spectral theory

DOI: 10.3233/ASY-2009-0919

Citation: Asymptotic Analysis, vol. 62, no. 3-4, pp. 163-205, 2009

Authors: Omel'chenko, Oleh | Recke, Lutz

Article Type: Research Article

Abstract: We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type ε2 u″=f(x, u, εu′, ε), 0<x<1, with Dirichlet and Neumann boundary conditions. For that we assume that there is given a family of approximate solutions which satisfy the differential equation and the boundary conditions with certain low accuracy. Moreover, we show that, if this accuracy is high, then the closeness of the approximate solution to the exact solution is correspondingly high. The main tool of the proofs is a generalized implicit function theorem which is close to those of Fife and …Greenlee (Uspechi Mat. Nauk 24 (1974), 103–130) and of Magnus (Proc. Royal Soc. Edinburgh 136A (2006), 559–583). Finally we show how to construct approximate solutions under certain natural conditions. Show more

Keywords: singular perturbation, asymptotic approximation, boundary layer, implicit function theorem

DOI: 10.3233/ASY-2009-0921

Citation: Asymptotic Analysis, vol. 62, no. 3-4, pp. 207-225, 2009

Authors: Gie, Gung-Min

Article Type: Research Article

Abstract: The goal of this article is to study the boundary layer of a reaction-diffusion equation with a small viscosity in a general (curved), bounded and smooth domain in Rn , n≥2. To the best of our knowledge, the classical expansion in the case of a bounded interval or of a channel is not valid for a general domain. Using the techniques of differential geometry, a new asymptotic expansion proposed in this article recovers the optimal convergence rate of the remainder at all orders.

Keywords: boundary layers, singular perturbation analysis, reaction–diffusion, curvilinear coordinates

DOI: 10.3233/ASY-2009-0922

Citation: Asymptotic Analysis, vol. 62, no. 3-4, pp. 227-249, 2009

Article Type: Other

Citation: Asymptotic Analysis, vol. 62, no. 3-4, pp. 251-251, 2009