In this paper, starting from classical non-convex and nonlocal 3-variational model of the electric polarization in a ferroelectric material, an asymptotic process we obtain a rigorous 2-variational model for a thin film. Depending on the initial boundary conditions, the limit problem can be either nonlocal or local.

We investigate the asymptotic behaviour, as $\epsilon \to 0$, of a class of monotone nonlinear Neumann problems, with growth $p-1$ ($p\in ]1,+\infty [$), on a bounded multidomain ${\Omega}_{\epsilon}\subset {\mathbb{R}}^{N}$
$(N\ge 2)$. The multidomain ${\Omega}_{\epsilon}$ is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness ${h}_{\epsilon}$ in the ${x}_{N}$ direction, as $\epsilon \to 0$. The second one is a “forest” of cylinders distributed with $\epsilon $-periodicity in the first $N-1$ directions on the upper side of the plate. Each cylinder has a small cross section of size $\epsilon $ and fixed...

We investigate the
asymptotic behaviour,
as ε → 0, of a class of monotone
nonlinear Neumann problems, with growth -1
( ∈]1, +∞[), on a bounded
multidomain
${\Omega}_{\epsilon}\subset {\mathbb{R}}^{N}$ ( ≥ 2). The multidomain
Ω is
composed of two domains. The first one
is a plate which becomes
asymptotically flat, with thickness
h in the
direction, as ε → 0.
The second one
is a “forest" of cylinders
distributed with
-periodicity in the first - 1 directions
on the upper side of the plate.
Each cylinder has
...

We study the asymptotic behaviour
of the following nonlinear problem:
$$\{\begin{array}{c}-\mathrm{div}\left(a\left(D{u}_{h}\right)\right)+{\left|{u}_{h}\right|}^{p-2}{u}_{h}=f\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}{\Omega}_{h},a\left(D{u}_{h}\right)\xb7\nu =0\phantom{\rule{1.0em}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}\partial {\Omega}_{h},\hfill \end{array}.$$
in a domain Ω_{
h
} of ${\mathbb{R}}^{n}$
whose boundary ∂Ω_{
h
}
contains an oscillating part with respect to h
when h tends to ∞.
The oscillating boundary is defined
by a set of cylinders with axis 0...

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses...

The paper is a continuation of a previous work of the same authors
dealing with homogenization processes for some energies
of integral type arising in the modeling of rubber-like elastomers.
The previous paper took into account the general case of the
homogenization of energies in presence of pointwise oscillating
constraints on the admissible deformations.
In the present paper homogenization processes are treated in the
particular case of fixed constraints set, in which minimal
coerciveness hypotheses...

We consider the linearized elasticity system in a multidomain of ${\mathbf{R}}^{3}$. This multidomain is the union of a horizontal plate with fixed cross section and small thickness ,
and of a vertical beam with fixed height and small cross section of radius ${r}^{\epsilon}$. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When and ${r}^{\epsilon}$ tend to zero simultaneously, with ${r}^{\epsilon}\gg {\epsilon}^{2}$, we identify the limit problem. This limit problem involves six junction conditions.

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