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### 3D-2D asymptotic analysis for micromagnetic thin films

ESAIM: Control, Optimisation and Calculus of Variations

$\Gamma$-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness $\epsilon$ approaches zero of a ferromagnetic thin structure ${\Omega }_{\epsilon }=\omega ×\left(-\epsilon ,\epsilon \right)$, $\omega \subset {ℝ}^{2}$, whose energy is given by${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W\left(\overline{m},\nabla \overline{m}\right)+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$subject to$\text{div}\left(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }}\right)=0\phantom{\rule{1.0em}{0ex}}\text{on}{ℝ}^{3},$and to the constraint$|\overline{m}|=1\text{on}{\Omega }_{\epsilon },$where $W$ is any continuous function satisfying $p$-growth assumptions with $p>1$. Partial results are also obtained in the case $p=1$, under an additional assumption on $W$.

### 3D-2D Asymptotic Analysis for Micromagnetic Thin Films

ESAIM: Control, Optimisation and Calculus of Variations

Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure ${\Omega }_{\epsilon }=\omega ×\left(-\epsilon ,\epsilon \right)$, $\omega \subset {ℝ}^{2}$, whose energy is given by ${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W\left(\overline{m},\nabla \overline{m}\right)+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$ subject to $\text{div}\left(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }}\right)=0\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}{ℝ}^{3},$ and to the constraint $|\overline{m}|=1\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}{\Omega }_{\epsilon },$ where W is any continuous function satisfying p-growth assumptions with p> 1. Partial results are also obtained in the case p=1, under an additional assumption on W.

### A new approach for estimating the friction in thin film lubrication.

Mathematical Problems in Engineering

### A two-dimensional Landau-Lifshitz model in studying thin film micromagnetics.

Abstract and Applied Analysis

### A waiting time phenomenon for thin film equations

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### Adhesion dynamics in probing micro- and nanoscale thin solid films.

Mathematical Problems in Engineering

### Asymptotic analysis, in a thin multidomain, of minimizing maps with values in ${S}^{2}$

Annales de l'I.H.P. Analyse non linéaire

### Effective energy integral functionals for thin films with bending moment in the Orlicz-Sobolev space setting

Banach Center Publications

In this paper we deal with the energy functionals for the elastic thin film ω ⊂ ℝ² involving the bending moments. The effective energy functional is obtained by Γ-convergence and 3D-2D dimension reduction techniques. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type and to satisfy the conditions Δ₂ and...

### Effective energy integral functionals for thin films with three dimensional bending moment in the Orlicz-Sobolev space setting

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider an elastic thin film ω ⊂ ℝ² with the bending moment depending also on the third thickness variable. The effective energy functional defined on the Orlicz-Sobolev space over ω is described by Γ-convergence and 3D-2D dimension reduction techniques. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type...

### Effects of In-plane Elastic Stress and Normal External Stress on Viscoelastic Thin Film Stability

Mathematical Modelling of Natural Phenomena

Motivated by recent experiments on the electro-hydrodynamic instability of spin-cast polymer films, we study the undulation instability of a thin viscoelastic polymer film under in-plane stress and in the presence of either a close by contactor or an electric field, both inducing a normal stress on the film surface. We find that the in-plane stress affects both the typical timescale of the instability and the unstable wavelengths. The film stability...

### Epitaxially strained elastic films: the case of anisotropic surface energies

ESAIM: Control, Optimisation and Calculus of Variations

In the context of a variational model for the epitaxial growth of strained elastic films, we study the effects of the presence of anisotropic surface energies in the determination of equilibrium configurations. We show that the threshold effect that describes the stability of flat morphologies in the isotropic case remains valid for weak anisotropies, but is no longer present in the case of highly anisotropic surface energies, where we show that the flat configuration is always a local minimizer...

### Equi-integrability results for 3D-2D dimension reduction problems

ESAIM: Control, Optimisation and Calculus of Variations

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left({\nabla }_{\alpha }{u}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_{3}{u}_{\epsilon }\right)$ bounded in ${L}^{p}\left(\Omega ;{ℝ}^{9}\right),\phantom{\rule{4pt}{0ex}}1<p<+\infty .$ Here it is shown that, up to a subsequence, ${u}_{\epsilon }$ may be decomposed as ${w}_{\epsilon }+{z}_{\epsilon },$ where ${z}_{\epsilon }$ carries all the concentration effects, i.e. $\left\{{\left|\left({\nabla }_{\alpha }{w}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_{3}{w}_{\epsilon }\right)\right|}^{p}\right\}$ is equi-integrable, and ${w}_{\epsilon }$ captures the oscillatory behavior, i.e. ${z}_{\epsilon }\to 0$ in measure. In addition, if $\left\{{u}_{\epsilon }\right\}$ is a recovering sequence then ${z}_{\epsilon }={z}_{\epsilon }\left({x}_{\alpha }\right)$ nearby $\partial \Omega .$

### Equi-integrability results for 3D-2D dimension reduction problems

ESAIM: Control, Optimisation and Calculus of Variations

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left({\nabla }_{\alpha }{u}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_{3}{u}_{\epsilon }\right)$ bounded in ${L}^{p}\left(\Omega ;{ℝ}^{9}\right),\phantom{\rule{4pt}{0ex}}1 Here it is shown that, up to a subsequence, ${u}_{\epsilon }$ may be decomposed as ${w}_{\epsilon }+{z}_{\epsilon },$ where ${z}_{\epsilon }$ carries all the concentration effects, i.e.$\left\{{\left|\left({\nabla }_{\alpha }{w}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_{3}{w}_{\epsilon }\right)\right|}^{p}\right\}$ is equi-integrable, and ${w}_{\epsilon }$ captures the oscillatory behavior, i.e.${z}_{\epsilon }\to 0$ in measure. In addition, if $\left\{{u}_{\epsilon }\right\}$ is a recovering sequence then ${z}_{\epsilon }={z}_{\epsilon }\left({x}_{\alpha }\right)$ nearby $\partial \Omega .$

### Heteroclinic orbits, mobility parameters and stability for thin film type equations.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### On the numerical modeling of deformations of pressurized martensitic thin films

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite.

### On the Numerical Modeling of Deformations of Pressurized Martensitic Thin Films

ESAIM: Mathematical Modelling and Numerical Analysis

We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite.

### Qualitative properties of a continuum theory for thin films

Annales de l'I.H.P. Analyse non linéaire

### Spatial heterogeneity in 3D-2D dimensional reduction

ESAIM: Control, Optimisation and Calculus of Variations

A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem...

### Spatial heterogeneity in 3D-2D dimensional reduction

ESAIM: Control, Optimisation and Calculus of Variations

A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem...

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