Asymptotic Analysis - Volume 25, issue 3-4

Authors: Schneider, Klaus R. | Vasil'eva, Adelaida B.

Article Type: Research Article

Abstract: We investigate steady state solutions to a class of systems of reaction–diffusion–convection equations with small diffusion and small convection in case of a scalar spatial variable. Our main concern is to prove the existence of a solution with an interior layer of spike type for higher order systems without taking into consideration the influence of boundary conditions. To this end we combine two methods of the theory of singular perturbations: the method of integral manifolds and the method of boundary layer functions.

Keywords: reaction–diffusion–convection equations, contrast structures of spike type, singular perturbations, asymptotic expansions, boundary layer functions, integral manifolds

Citation: Asymptotic Analysis, vol. 25, no. 3-4, pp. 183-199, 2001

Authors: Grébert, B. | Kappeler, T.

Article Type: Research Article

Abstract: Consider the 2×2 first order system due to Zakharov–Shabat, LY:={\rm i}\left(\matrix{1&0\cr 0&-1}\right)Y'+\left(\matrix{0&\psi_{1}\cr \psi _{2}&0}\right)Y=\lambda Y with ψ1 ,ψ2 being complex valued functions of period one in the weighted Sobolev space $H^{w}\equiv H^{w}_{\mathbb{C}}.$ Denote by spec(ψ1 ,ψ2 ) the set of periodic eigenvalues of L(ψ1 ,ψ2 ) with respect to the interval [0,2] and by specDir (ψ1 ,ψ2 ) the set of Dirichlet eigenvalues of L(ψ1 ,ψ2 ) when considered on the interval [0,1]. It is well known that spec(ψ1 ,ψ2 ) and specDir (ψ1 ,ψ2 ) are discrete. Theorem. Assume that w is a …weight such that, for some δ>0, w−δ (k)=(1+|k|)−δ w(k) is a weight as well. Then for any bounded subset ${\mathbb{B}}$ of 1‐periodic elements in Hw ×Hw there exist N≥1 and M≥1 so that for any |k|≥N, and $(\psi_{1},\psi_{2})\in{\mathbb{B}} $ , the set $\mathit{spec}(\psi_{1},\psi_{2})\cap \{\lambda \in{\mathbb{C}} \mid |\lambda -k\pi | < \pi/2\}$ contains exactly one isolated pair of eigenvalues {λ+ k ,λ− k } and $\mathit{spec}_{\rm Dir}(\psi_{1},\psi_{2})\cap \{\lambda \in{\mathbb{C}} \mid |\lambda -k\pi |<{\pi}/{2}\}$ contains a single Dirichlet eigenvalue μk . These eigenvalues satisfy the following estimates (i) Σ|k|≥N w(2k)2 |λ+ k −λ− k |2 ≤M; (ii) $\sum _{|k|\geq N}w(2k)^{2}|\frac{(\lambda ^{+}_{k}+\lambda ^{-}_{k})} {2}-\mu _{k}|^{2}\leq M.$ Furthermore spec(ψ1 ,ψ2 )\{λ± k ,|k|≥N} and $\mathit{spec}_{\rm Dir}(\psi_{1},\psi_{2})\backslash \{\mu _{k}\mid |k|\geq N\}$ are contained in $\{\lambda \in{\mathbb{C}} \mid |\lambda | < N\pi -\pi /2\}$ and its cardinality is 4N−2, respectively 2N−1. When $\psi_{2}=\overline{\psi } _{1}$ (respectively $\psi_{2}=-\overline{\psi } _{1}),L(\psi_{1},\psi_{2})$ is one of the operators in the Lax pair for the defocusing (resp. focusing) nonlinear Schrödinger equation. Show more

Citation: Asymptotic Analysis, vol. 25, no. 3-4, pp. 201-237, 2001

Authors: Andreoiu, Georgiana | Faou, Erwan

Article Type: Research Article

Abstract: In this paper we study the asymptotics of the three‐dimensional displacement field for clamped and free linear elastic shallow shells as the thickness tends to zero. As in the case of plates, the asymptotics contains regular terms and boundary layers. The two‐dimensional generators of the regular parts are solutions of two‐dimensional problems governed by an elliptic system in the sense of S. Agmon, A. Douglis and L. Nirenberg. This asymptotics is justified by optimal error estimates and improves the results obtained by S. Busse, P.G. Ciarlet and B. Miara.

Keywords: mechanics of solids, shallow shells, linear elasticity, boundary layers

Citation: Asymptotic Analysis, vol. 25, no. 3-4, pp. 239-270, 2001

Authors: Briane, Marc

Article Type: Research Article

Abstract: This paper is devoted to the homogenization of the problem −div(aε ∇uε )+ν uε =f in a bounded domain Ω of Rd , with Neumann's (ν=1) or Dirichlet's (ν=0) boundary conditions. The conductivity matrix aε is defined by $a_{\varepsilon }(x):=A_{\varepsilon }(\tfrac{x}{\varepsilon })$ where (Aε )ε>0 is a sequence of bounded but non‐uniformly elliptic periodic matrix‐valued functions. We make a general assumption on Aε for that the sequence uε strongly converges in L2 (Ω) to a function u0 solution of a similar problem. We also yield an example in which the compactness result holds …true although the sequence Aε uniformly looses its ellipticity as ε tends to zero. Finally we illustrate the optimality of our condition on Aε in the framework of isolating thin layers. Show more

Citation: Asymptotic Analysis, vol. 25, no. 3-4, pp. 271-297, 2001

Authors: Bochniak, M. | Sändig, A.‐M.

Article Type: Research Article

Abstract: Material discontinuities and geometrical peculiarities like cracks, notches, corners and edges lead to stress singularities in linear elastic structures. Their strength can be characterised by stress intensity or notch factors. In this paper we investigate the influence of the shape and of the material parameters on these factors. The sensitivity analysis is performed by means of asymptotic expansions.

Citation: Asymptotic Analysis, vol. 25, no. 3-4, pp. 299-328, 2001

Authors: Bayada, G. | Lhalouani, K.

Article Type: Research Article

Abstract: On étudie le comportement asymptotique d'une structure à 3 corps constituée d'une couche mince élastique placée entre un corps rigide et un corps élastique. On suppose que le contact entre les deux corps élastique est un contact unilatéral avec frottement de Coulomb non local et que les coefficients de Lamé de la couche mince dépendent de son épaisseur ε. On montre que lorsque l'épaisseur du joint tend vers zéro la structure initiale peut être remplacée par une structure à deux corps avec de nouvelles conditions à l'interface. Enfin, on présente quelques simulations numériques permettant de mettre en évidence les résultats …obtenus théoriquement. Show more

Citation: Asymptotic Analysis, vol. 25, no. 3-4, pp. 329-362, 2001

Article Type: Other

Citation: Asymptotic Analysis, vol. 25, no. 3-4, pp. 363-363, 2001