This paper is devoted to the study of the linear parabolic problem $\epsilon {\partial }_t{u}_{\epsilon }(x,t)-\nabla ·\left(a(x/\epsilon ,t/{\epsilon }^3)\nabla u_{\epsilon }(x,t)\right)=f(x,t)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $\epsilon$ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic local problems....

We study the asymptotic behaviour of the following nonlinear problem: $$\{\begin{array}{c}-\mathrm{div}\left(a\left(D{u}_h\right)\right)+{\left|u_h\right|}^{p-2}{u}_h=f\text{in}{\Omega }_h,a\left(D{u}_h\right)·\nu =0\text{on}\partial {\Omega }_h,\hfill \end{array}.$$ in a domain Ωh of ${ℝ}^n$ whose boundary ∂Ωh contains an oscillating part with respect to h when h tends to ∞. The oscillating boundary is defined by a set of cylinders with axis 0xn that are h-1-periodically distributed. We prove that the limit problem in the domain corresponding to the oscillating boundary identifies with a diffusion operator with respect to xn coupled with an algebraic problem for the limit fluxes.

In L2(ℝd; ℂn), we consider a wide class of matrix elliptic second order differential operators $𝒜$ε with rapidly oscillating coefficients (depending on x/ε). For a fixed τ > 0 and small ε > 0, we find approximation of the operator exponential exp(− $𝒜$ετ) in the (L2(ℝd; ℂn) → H1(ℝd; ℂn))-operator norm with an error term of order ε. In this approximation, the corrector is taken...

In this paper, a singular semi-linear parabolic PDE with locally periodic coefficients is homogenized. We substantially weaken previous assumptions on the coefficients. In particular, we prove new ergodic theorems. We show that in such a weak setting on the coefficients, the proper statement of the homogenization property concerns viscosity solutions, though we need a bounded Lipschitz terminal condition.

This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the said zero order terms, the family of solutions of the studied problem converges in law, and describe the limit process. It should be noted that the limit dynamics remain random.

This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two...

This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product...

In this work we consider a diffusion problem in a periodic composite having three phases: matrix, fibers and interphase. The heat conductivities of the medium vary periodically with a period of size ${\epsilon }^{\beta }$ ($\epsilon >0$ and $\beta >0$) in the transverse directions of the fibers. In addition, we assume that the conductivity of the interphase material and the anisotropy contrast of the material in the fibers are of the same order ${\epsilon }^2$ (the so-called double-porosity type scaling) while the matrix material has a conductivity of...

We are concerned with the asymptotic analysis of optimal control problems for $1$-D partial differential equations defined on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a two-dimensional homogenized system, and...

We are concerned with the asymptotic analysis of optimal control problems for 1-D partial differential equations defined on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a two-dimensional homogenized system,...

The paper deals with a scalar diffusion equation $c{u}_t={\left(F\left[u_x\right]\right)}_x+f,$ where $F$ is a Prandtl-Ishlinskii operator and $c,f$ are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary value problem. The problem is then homogenized by considering a sequence of equations of the above type with spatially periodic...