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The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.
Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.
Authors: Barles, Guy | Mironescu, Elisabeth
Article Type: Research Article
Abstract: We study two types of asymptotic problems whose common feature – and difficulty – is to exhibit oscillating Dirichlet boundary conditions: the main contribution of this article is to show how to recover the Dirichlet boundary condition for the limiting equation. These two types of problems are (i) periodic homogenization problems for fully nonlinear, second-order elliptic partial differential equations set in a half-space and (ii) parabolic problems with an oscillating in time Dirichlet boundary condition. In order to obtain the Dirichlet boundary condition for the limiting problem, the key step is a blow-up argument near the boundary which leads to …the study of Dirichlet problems set on half-space type domains and of the asymptotic behavior of the solutions when the distance to the boundary tends to infinity. Show more
Keywords: homogenization, oscillating Dirichlet boundary conditions, fully nonlinear elliptic equations, viscosity solutions
DOI: 10.3233/ASY-2012-1139
Citation: Asymptotic Analysis, vol. 82, no. 3-4, pp. 187-200, 2013
Authors: Cioranescu, Doina | Damlamian, Alain | Orlik, Julia
Article Type: Research Article
Abstract: We consider the elasticity problem in a heterogeneous domain with an ε-periodic micro-structure, ε<<1, including multiple micro-contacts between the structural components. These components can be a simply connected matrix domain with open cracks or inclusions completely surrounded by cracks, which do not touch the outer boundary. The contacts are described by the Signorini and Tresca-friction contact conditions. The Signorini condition is described mathematically by a closed convex cone, while the friction condition is a nonlinear convex functional over the interface jump of the solution on the oscillating interface. The difficulties appear when the inclusions are completely surrounded by cracks …and can have rigid displacements. In this case, in order to obtain preliminary estimates for the solution in the ε-domain, the Korn inequality should be modified, first in the fixed context and then for the ε-dependent periodic case. Additionally, for all states of the contact (inclusions can freely move, or are locked/sticked to the interface with the matrix, or the frictional traction is achieved on the inclusion-matrix interface and the inclusions can slide in the tangential to the interface direction) we obtain estimates for the solution in the ε-domain, uniform with respect to ε. An asymptotic analysis (as ε→0) for the nonlinear functionals over the growing interface is carried out, based on the application of the periodic unfolding method for sequences of jumps of the solution on the oscillating interface. This allows to obtain the homogenized limit as well as a corrector result. Show more
Keywords: Korn inequality for contact with inclusions, unfolding on the oscillating interface, convergence of traces, jumps and some non-linear Robin-type conditions, homogenization of contact
DOI: 10.3233/ASY-2012-1141
Citation: Asymptotic Analysis, vol. 82, no. 3-4, pp. 201-232, 2013
Authors: Visintin, Augusto
Article Type: Research Article
Abstract: In homogenization, two-scale models arise, e.g., by applying Nguetseng's notion of two-scale convergence to nonlinear PDEs. A homogenized single-scale problem may then be derived via scale-transformations. A variational formulation due to Fitzpatrick is here used for the scale-integration of two-scale maximal monotone relations, and for the converse operation of scale-disintegration. These results are applied to the periodic homogenization of a quasilinear model of Ohmic electric conduction with Hall effect: $\lefteqn{\vec{E}\in\vec{\alpha}(\vec{J},x/\varepsilon )+h(x/\varepsilon )\vec{J}\times\vec{B}(x/\varepsilon )+\vec{E}_{a}(x/\varepsilon ),}$ $\lefteqn{\nabla\times\vec{E}=\vec{g}(x/\varepsilon ),\qquad\nabla\cdot\vec{J}=0\quad\mbox{in }\varOmega,}$ with $\vec{\alpha}(\cdot,x/\varepsilon )$ maximal monotone, $\vec{B},\vec{E}_{a},h,\vec{g}$ prescribed fields. (This corresponds to a …quasilinear second-order elliptic equation in curl form: $\nabla\times\vec{\beta}(\nabla\times u,x/\varepsilon )=\vec{g}(x/\varepsilon )$ .) This result is also retrieved via De Giorgi's Γ-convergence. Show more
Keywords: scale-transformations, monotone operators, homogenization, two-scale convergence, Ohm and Hall laws, Γ-convergence
DOI: 10.3233/ASY-2012-1143
Citation: Asymptotic Analysis, vol. 82, no. 3-4, pp. 233-270, 2013
Authors: Waurick, Marcus
Article Type: Research Article
Abstract: We present a unified Hilbert space perspective to homogenization of a class of evolutionary equations of mathematical physics. We formulate homogenization in a purely operator-theoretic setting. Using “A structural observation for linear material laws in classical mathematical physics” by R. Picard [Mathematical Methods in the Applied Sciences 32 (2009), 1768–1803], we discuss constitutive relations as certain elements of the Hardy space ℋ∞ (E;L(H)) of bounded, analytic and operator-valued functions M:E→L(H), where E\subseteqq C open, H Hilbert space. The core idea is to introduce a certain topology on the set of constitutive relations. Given a convergent sequence of constitutive relations, the …behavior of solutions to the respective problems is discussed. We apply the results to the equations of acoustics, thermodynamics, elasticity or coupled systems such as thermo-elasticity. The respective equations may also incorporate memory or delay terms and fractional derivatives. In particular, constitutive relations via differential equations can also be treated. Show more
Keywords: homogenization, evolutionary differential equations, delay and memory effects, fractional derivatives, thermo-elasticity
DOI: 10.3233/ASY-2012-1145
Citation: Asymptotic Analysis, vol. 82, no. 3-4, pp. 271-294, 2013
Authors: Tersenov, Alkis S.
Article Type: Research Article
Abstract: In the present paper we study the Cauchy problem as well as the first initial boundary value problem for a class of quasilinear ultraparabolic equations. We show that the presence of the low order term satisfying a certain assumption provides a global solvability of the above problems. The optimality of this assumption is demonstrated.
Keywords: partial diffusivity, a priori estimates, global solvability
DOI: 10.3233/ASY-2012-1146
Citation: Asymptotic Analysis, vol. 82, no. 3-4, pp. 295-314, 2013
Authors: Berbiche, Mohamed
Article Type: Research Article
Abstract: In this paper we consider the Cauchy problem of semi-linear damped wave equation with nonlinear memory term a∫0 t (t−τ)−γ |u|α−1 u(τ,x) dτ. We prove global existence and asymptotic behavior of solution for small initial data. Moreover, we show that the global solutions behave asymptotically like self-similar solutions of the semi-linear heat equation with nonlinear memory as t→∞.
Keywords: damped wave equation, self-similar solutions, semi-linear heat equation
DOI: 10.3233/ASY-2012-1147
Citation: Asymptotic Analysis, vol. 82, no. 3-4, pp. 315-330, 2013
Authors: Takeda, Hiroshi | Yoshikawa, Shuji
Article Type: Research Article
Abstract: The aim of this paper is to investigate asymptotic behavior of solutions to the Cauchy problem of certain fourth-order semilinear elastic system with weak damping in three space dimension. The system corresponds to the isothermal case of the thermoelastic system representing the martensitic phase transitions on shape memory alloys. We give several asymptotic profiles of solutions. By considering the second-order expansion of solutions, we obtain more precise information about asymptotic behavior of solutions as t→∞, and observe the contribution of the nonlinear term to solutions. Moreover, we give the third-order expansion formula which enable us to clarify the relation between …the fourth- and the second-order derivative term. Show more
Keywords: asymptotic profile, shape memory alloy, initial value problem, semilinear elastic system, higher-order expansion of solution
DOI: 10.3233/ASY-2012-1148
Citation: Asymptotic Analysis, vol. 82, no. 3-4, pp. 331-372, 2013
Article Type: Other
Citation: Asymptotic Analysis, vol. 82, no. 3-4, pp. 373-374, 2013
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