Asymptotic Analysis - Volume 44, issue 3-4

Authors: Immink, G.K.

Article Type: Research Article

Abstract: We prove an asymptotic existence theorem for locally analytic, nonlinear difference equations possessing a formal power series solution at ∞. The proof is based on a well-known contraction argument applied to Banach spaces of holomorphic functions on suitable unbounded domains of \[$\mathbb{C}$ . The main difficulty lies in the definition of appropriate domains for the case that the equation possesses a level “1+ ” in combination with lower levels.

Citation: Asymptotic Analysis, vol. 44, no. 3-4, pp. 173-220, 2005

Authors: Carozza, Menita | Moscariello, Gioconda | Passarelli di Napoli, Antonia

Article Type: Research Article

Abstract: We prove a regularity result for local minimizers of degenerate variational integrals, whose model arises in the study of mappings with finite distortion. The degeneracy function 𝒦(x) lies in the exponential class, i.e., exp (λ𝒦(x)) is integrable for some λ>0. The right space of the gradient of a local minimizer u turns out to be the Zygmund class Lp log −1 L. Our result states that if λ is sufficiently large, then Du belongs to the Zygmund space Lp log α L, α≥1 and α increases with λ.

Keywords: degenerate variational integrals, duality theory, mappings with finite distortion

Citation: Asymptotic Analysis, vol. 44, no. 3-4, pp. 221-235, 2005

Authors: Coron, J.-M. | Guerrero, S.

Article Type: Research Article

Abstract: We consider the problem of the null controllability of a family of 1-D linear control parabolic equations depending on two parameters, namely the viscosity and the coefficient of the transport term. We study the dependence, with respect to these parameters and the time of controllability, of the norm of the optimal controls. In particular we give estimates on the optimal control as the viscosity tends to 0.

Citation: Asymptotic Analysis, vol. 44, no. 3-4, pp. 237-257, 2005

Authors: Nazarov, Sergueï A. | Thäter, Gudrun

Article Type: Research Article

Abstract: The objective of our paper is to construct the asymptotic representation of solutions to our problem at infinity. In doing so the main difficulty is the justification of the formal procedure presented at the beginning. Then, based on the asymptotics, we calculate integral representations for the involved coefficients and prove the Fredholm property of our problem in step-weighted spaces providing at the same moment kernel and cokernel.

Citation: Asymptotic Analysis, vol. 44, no. 3-4, pp. 259-298, 2005

Authors: Fonseca, Irene | Popovici, Cristina

Article Type: Research Article

Abstract: The \[$\varGamma(L^{1}(\varOmega;\mathbb{R}^{d}))$ -limit of the sequence \[J_{\varepsilon}(u):=\frac{1}{\varepsilon}E_{\varepsilon}(u),\] where Eε is the family of anisotropic singular perturbations Eε (u):=∫Ω f(x,u(x),ε∇u(x)) dx of a non-convex functional of vector-valued functions E(u):=∫Ω f(x,u(x),∇u(x)) dx is obtained where f is a non-negative energy density satisfying f(x,u,0)=0 if and only if u∈{a,b}.

Keywords: \[$\varGamma$-convergence, phase transitions, singular perturbations, double-well potential

Citation: Asymptotic Analysis, vol. 44, no. 3-4, pp. 299-325, 2005

Authors: Figueiredo, Isabel M. Narra | Leal, Carlos M. Franco

Article Type: Research Article

Abstract: We mathematically justify a reduced piezoelectric plate model. This is achieved considering the three-dimensional static equations of piezoelectricity, for a nonhomogeneous anisotropic thin plate, and using the asymptotic analysis to compute the limit of the displacement vector and electric potential, as the thickness of the plate approaches zero. We prove that the three-dimensional displacement vector converges to a Kirchhoff–Love displacement, that solves a two-dimensional piezoelectric plate model, defined on the middle surface of the plate. Moreover, the three-dimensional electric potential converges to a scalar function that is a second-order polynomial with respect to the thickness variable, with coefficients that depend …on the transverse component of the Kirchhoff–Love displacement. We remark that the results of this paper generalize a previous work of A. Sene (Asymptotic Anal. 25(1) (2001), 1–20) for homogeneous and isotropic materials. Show more

Keywords: asymptotic analysis, anisotropic material, piezoelectricity, plate

Citation: Asymptotic Analysis, vol. 44, no. 3-4, pp. 327-346, 2005

Article Type: Other

Citation: Asymptotic Analysis, vol. 44, no. 3-4, pp. 347-348, 2005