Asymptotic Analysis - Volume 38, issue 3-4

Authors: Yi, Fahuai | Wang, Lijie

Article Type: Research Article

Abstract: A free boundary problem for a parabolic system arising from the mathematical theory of combustion in multidimensional case will be considered in this paper. The existence and uniqueness of classical solution locally in time will be obtained by the use of contraction mapping principle. A convergence result for parameter λ→0 is also obtained.

Keywords: free boundary problem, combustion, parabolic system

Citation: Asymptotic Analysis, vol. 38, no. 3-4, pp. 187-199, 2004

Authors: Cardone, G. | Corbo Esposito, A. | Yosifian, G.A. | Zhikov, V.V.

Article Type: Research Article

Abstract: A homogenization theorem is established for the problem of minimization of a quadratic integral functional on a set of admissible functions whose gradients are subjected to rapidly changing constraints imposed on a disperse periodic set (regarded as inclusions). At each point of the inclusion, the gradients must belong to a given closed convex set of an arbitrary structure which may vary from point to point within the inclusion. Our approach is based on two‐scale convergence and an explicit construction of a Γ‐realizing sequence. This homogenization method can be directly applied to variational problems for vector‐valued functions, which is demonstrated on …problems of elasticity with convex constraints on the strain tensor at the points of disperse inclusions. We also consider some problems with constraints on periodic sets of zero Lebesgue measure and study homogenization problems for some cases of nondisperse inclusions. Show more

Keywords: gradient constraints, homogenization, two‐scale convergence, Γ‐realizing sequence, elasticity

Citation: Asymptotic Analysis, vol. 38, no. 3-4, pp. 201-220, 2004

Authors: Zappacosta, Stefano

Article Type: Research Article

Abstract: We improve estimates for power resolvent of the free Laplacian −Δ on $\mathbb{R}^{n}$ , and we use them to get a control on the solution to the wave equation with a potential (∂2 t −Δ)u+Vu=0 in $\mathbb{R}^{3}$ . We finally obtain a local energy decay for u.

Keywords: wave equations, power resolvent estimates, local energy decay

Citation: Asymptotic Analysis, vol. 38, no. 3-4, pp. 221-239, 2004

Authors: Khenissy, Saïma

Article Type: Research Article

Abstract: We consider the minimization of the Dirichlet integral ∫Ω |∇u|2 with the constraint ∫Ω (1−|u|2 )2 ≤λ, for maps u∈H1 (Ω;$\mathbb{R} ^{2})$ , where Ω⊂$\mathbb{R} ^{2}$ is a smooth, bounded and simply connected domain, u=g on ∂Ω with g :∂Ω→S1 unit circle in $\mathbb{R} ^{2}$ , and λ is a positive small parameter. Denoting by d the topological degree of g, we study the asymptotic behavior of a minimizer uλ when λ goes to zero, for d=0 and d≠0. When d=0, we show the convergence of uλ in various norms to a smooth …harmonic map u*  :Ω→S1 . When d≠0 (d>0) we show that uλ converges to a smooth harmonic map u0  :Ω\{a1 ,a2 ,…,ad }→S1 , where a1 ,a2 ,…,ad are the vortices where the energy of a minimizer ∫Ω |∇uλ |2 concentrates, when λ→0. Show more

Citation: Asymptotic Analysis, vol. 38, no. 3-4, pp. 241-291, 2004

Authors: Molle, Riccardo

Article Type: Research Article

Abstract: This paper deals with a class of singularly perturbed nonlinear elliptic problems (Pε ) with subcritical nonlinearity. The coefficient of the linear part is assumed to concentrate in a point of the domain, as ε→0, and the domain is supposed to be unbounded and with unbounded boundary. Domains that enlarge at infinity, and whose boundary flattens or shrinks at infinity, are considered. It is proved that in such domains problem (Pε ) has at least 2 solutions.

Keywords: unbounded domains, unbounded boundary, concentrating potential, multiple solutions

Citation: Asymptotic Analysis, vol. 38, no. 3-4, pp. 293-307, 2004

Authors: Grasman, Johan | Shih, Shagi‐Di

Article Type: Research Article

Abstract: A method is presented to approximate with singular perturbation methods a parabolic differential equation for the quarter plane with a discontinuity at the corner. This discontinuity gives rise to an internal layer. It is necessary to match the local solution in this layer with the one in a corner layer as otherwise terms in the internal layer solution remain unnoticed. The problem is explained using the exact solution of a special case. The asymptotic solution is proved to approximate the exact solution in the general case using the maximum principle for parabolic differential equations.

Keywords: singular perturbation, asymptotic approximation, internal layer, corner layer, asymptotic matching, maximum principle

Citation: Asymptotic Analysis, vol. 38, no. 3-4, pp. 309-318, 2004

Authors: Rahmani, Leila

Article Type: Research Article

Abstract: In this paper, we consider the full system of dynamic von Karman equations for an heterogeneous plate that comprises two parts: a thin rigid body inserted into an elastic plate. We show that Ventcel's boundary conditions may be obtained, as the thickness of the rigid body goes to zero.

Keywords: Ventcel's conditions, full von Karman system, multi‐structures, nonlinear plate, stiffener, asymptotic analysis

Citation: Asymptotic Analysis, vol. 38, no. 3-4, pp. 319-337, 2004

Authors: Chae, Dongho

Article Type: Research Article

Abstract: We prove the local in time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler equations of inviscid incompressible fluid flows in Rn , n≥2. As a corollary we obtain the persistence of Besov space regularity for the solutions of the 2‐D Euler equations with initial velocity belonging to the Besov spaces. For the proof of the results we establish a logarithmic inequality of the Beale–Kato–Majda type and a Moser type of inequality in the Besov spaces.

Citation: Asymptotic Analysis, vol. 38, no. 3-4, pp. 339-358, 2004