### 1 Structure de contact et équations aux dérivées partielles d'après V. V. Lychagin

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$\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 ${\Omega}_{\epsilon}=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {\mathbb{R}}^{2}$, whose energy is given by$${\mathcal{E}}_{\epsilon}\left(\overline{m}\right)=\frac{1}{\epsilon}{\int}_{{\Omega}_{\epsilon}}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}\xb7\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$$subject to$$\text{div}(-\nabla \overline{u}+\overline{m}{\chi}_{{\Omega}_{\epsilon}})=0\phantom{\rule{1.0em}{0ex}}\text{on}{\mathbb{R}}^{3},$$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 $W$.

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 ${\Omega}_{\epsilon}=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {\mathbb{R}}^{2}$, whose energy is given by $${\mathcal{E}}_{\epsilon}\left(\overline{m}\right)=\frac{1}{\epsilon}{\int}_{{\Omega}_{\epsilon}}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}\xb7\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$$ subject to $$\text{div}(-\nabla \overline{u}+\overline{m}{\chi}_{{\Omega}_{\epsilon}})=0\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}{\mathbb{R}}^{3},$$ 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.

In this paper, we introduce a set of methods for processing and analyzing long time series of 3D images representing embryo evolution. The images are obtained by in vivo scanning using a confocal microscope where one of the channels represents the cell nuclei and the other one the cell membranes. Our image processing chain consists of three steps: image filtering, object counting (center detection) and segmentation. The corresponding methods are based on numerical solution of nonlinear PDEs, namely...

In this paper a mixed boundary value problem for the fourth order hyperbolic equation with constant coefficients which is connected with response of semi-space to a short laser pulse» and belongs to generalized Thermoelasticity is studied. This problem was considered by R. B. Hetnarski and J. Ignaczak, who established some important physical consequences. The present paper contains proof of the existence, uniqueness and continuous dependence of a solution on the datum, together with an effective...

Given a domain $\Omega $ of class ${C}^{k,1}$, $k\in \mathbb{N}$, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of $\Omega $ in the sense that $(\partial -\partial {x}_{n})\alpha ({x}^{\text{'}},0)=-N\left({x}^{\text{'}}\right)$ and that still is of class ${C}^{k,1}$. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to $k$ on domains of class ${C}^{k,1}$. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.

We address the issue of parameter variations in POD approximations of time-dependent problems, without any specific restriction on the form of parameter dependence. Considering a parabolic model problem, we propose a POD construction strategy allowing us to obtain some a priori error estimates controlled by the POD remainder – in the construction procedure – and some parameter-wise interpolation errors for the model solutions. We provide a thorough numerical assessment of this strategy with the...