Asymptotic Analysis

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.

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.

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.

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.

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.

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−1u(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→∞.

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.