Asymptotic Analysis - Volume 62, issue 1-2

Authors: Wellander, Niklas

Article Type: Research Article

Abstract: A two-scale Fourier transform for periodic homogenization in Fourier space is introduced. The transform connects the various existing techniques for periodic homogenization, i.e., two-scale convergence, periodic unfolding and the Floquet–Bloch expansion approach to homogenization. It turns out that the two-scale compactness results are easily obtained by the use of the two-scale Fourier transform. Moreover, the Floquet–Bloch eigenvalue problems for differential operators is recovered in a natural and straight forward way by the use of this transform. The transform is generalized to the (N+1)-scale case.

Keywords: two-scale Fourier transform, two-scale convergence, weak convergence, homogenization, two-scale transform, periodic unfolding, periodic extension, Floquet–Bloch homogenization

DOI: 10.3233/ASY-2008-0914

Citation: Asymptotic Analysis, vol. 62, no. 1-2, pp. 1-40, 2009

Authors: Cardone, G. | Nazarov, S.A. | Sokolowski, J.

Article Type: Research Article

Abstract: The Neumann problem for the Poisson equation is considered in a domain Ωε ⊂Rn with boundary components posed at a small distance ε>0 so that in the limit, as ε→0+ , the components touch each other at the point 𝒪 with the tangency exponent 2m≥2. Asymptotics of the solution uε and the Dirichlet integral ‖∇x uε ; L2 (Ωε )‖2 are evaluated and it is shown that main asymptotic term of uε and the existence of the finite limit of the integral depend on the relation between the spatial dimension n and the exponent 2m. For …example, in the case n<2m−1 the main asymptotic term becomes of the boundary layer type and the Dirichlet integral has no finite limit. Some generalizations are discussed and certain unsolved problems are formulated, in particular, non-integer exponents 2m and tangency of the boundary components along smooth curves. Show more

Keywords: singularly perturbed Neumann problem, Dirichlet integral, touching surfaces, thin ligament

DOI: 10.3233/ASY-2008-0915

Citation: Asymptotic Analysis, vol. 62, no. 1-2, pp. 41-88, 2009

Authors: Fila, Marek | Winkler, Michael

Article Type: Research Article

Abstract: We analyze a mathematical model introduced by Anguige, Ward and King (J. Math. Biol. 51 (2005), 557–594) to describe a quorum sensing mechanism in spatially-structured populations of the bacteria Pseudomonas aeruginosa. In the biologically relevant limit case when the spatial distribution of bacteria is constant in time, this model reduces to a single semilinear parabolic equation for the concentration of the signal substance N-(3-oxododecanoyl)-homoserine lactone (AHL). We show that under some mild technical assumptions on the nonlinear AHL production rate, the AHL concentration approaches a uniquely determined steady state, and that this convergence takes place at an exponential rate.

Keywords: semilinear parabolic equation, quorum sensing, steady state, stabilization

DOI: 10.3233/ASY-2008-0920

Citation: Asymptotic Analysis, vol. 62, no. 1-2, pp. 89-106, 2009

Authors: Kałamajska, Agnieszka

Article Type: Research Article

Abstract: Let Ω⊆Rn be a bounded open domain, F⊆Omega¯ be a closed subset and let {uk }k∈N be a bounded sequence in Sobolew space where 1≤p<∞, converging weakly to u∈W1,p (Ω, Rm ). Let ℛ be a given complete separable ring of continuous functions on Rm×n and assume that for each f∈ℛ the sequence of compositions {f(∇uk )(1+|∇uk |p ) dx}k∈N embedded into the space of measures on Ω¯ converges weakly * to some measure μf . We discuss the possibility to modify the sequence {uk }k∈N in such a way that the new sequence {wk }k∈N …is still bounded in W1,p (Ω, Rm ), converges weakly to u, each sequence of measures {f(∇wk )(1+|∇wk |p ) dx}k∈N also converges weakly * to μf , where f∈ℛ, but additionally the new sequence satisfies the condition “wk =u” on F. Our results are applied to the minimization problems in the Calculus of Variations. Show more

Keywords: sequences of gradients, DiPerna–Majda measures, concentrations, oscillations

DOI: 10.3233/ASY-2008-0917

Citation: Asymptotic Analysis, vol. 62, no. 1-2, pp. 107-123, 2009