
Patrizia Donato Université de Rouen
Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and $L^1$ data

Antoine Gloria Université Pierre et Marie Curie (UPMC) Paris 6
The homogenization commutator and the pathwise structure of fluctuations
In this talk I will introduce the notion of homogenization commutator. The fluctuations at large scales of this stationary field characterize in a pathwise sense the fluctuations at large scales of standard quantities like the flux and the gradient of the corrector, and of the solution operator. In the first part of the talk, I will address this pathwise structure in two typical settings: Gaussian coefficients with integrable correlations and Gaussian coefficients with nonintegrable correlations. In the second part of the talk, I will describe the large scale fluctuations of the homogenization commutator: White noise for integrable correlations, and fractional white noise for nonintegrable correlations. This is based on joint works with M. Duerinckx, J. Fischer, S. Neukamm, and F. Otto. 
Georges Griso Université Pierre et Marie Curie (UPMC) Paris 6
Homogenization of a thin elastic plate
Find $ u_{\epsilon,\delta}$ in $\mathbf H_{\Gamma_{0,\delta}}(\Omega_\delta)\doteq\big\{ v\in H^1(\Omega_\delta;\mathbb R^3)$ so that $ v=0$ on $\Gamma_{0,\delta}\doteq \partial \omega\times (\delta,\delta)\big\}$, satisfying
$ \int_{\Omega_\delta} a_{ijkl}\big({x_1\over \epsilon}, {x_2\over \epsilon}, {x_3\over \delta}\big) \gamma_{ij}(u_{\epsilon,\delta})\gamma_{kl}(v)\, dx=\int_{\Omega_\delta} f_{\delta}(x)\, v(x)\, dx$ for all $ v\in {\mathbf H}_{\Gamma_{0,\delta}}(\Omega_\delta) $, where $a_{ijkl}\in L^\infty \big(Y\times (1,1)\big)$, $Y\doteq(0,1)^2$ and satisfy the usual conditions,
 $a_{ijkl}$ $Y$periodic,
 $\gamma_{ij}(v)=\frac12 \Big({\partial v_i\over \partial x_j}+{\partial v_j\over \partial x_i}\Big)$.
 1: $\epsilon\to 0$ ($\delta$ fixed) then $\delta\to 0$,
 2: $\delta\to 0$ ($\epsilon$ fixed) then $\epsilon\to 0$,
 3: $(\epsilon,\delta)\to (0,0)$ and $ \lim_{(\epsilon,\delta)\to (0,0)}{\epsilon/ \delta}=\theta\in [0,+\infty]$.

Ralf Kornhuber Free University Berlin
Towards numerical homogenization of multiscale fault networks

Agnes Lamacz Technical University Dortmund
Effective Maxwell's equations in a geometry with flat splitrings and wires
Propagation of light in heterogeneous media is a complex subject of research. It has received renewed interest in recent years, since technical progress allows for smaller devices and offers new possibilities. At the same time, theoretical ideas inspired further research. Key research areas are photonic crystals, negative index metamaterials, perfect imaging, and cloaking. The mathematical analysis of negative index materials, which we want to focus on in this talk, is connected to a study of singular limits in Maxwell's equations. We present a result on homogenization of the time harmonic Maxwell's equations in a complex geometry. The homogenization process is performed in the case that many (order $\eta^{3}$) small (order $\eta^1$), flat (order $\eta^2$) and highly conductive (order $\eta^{3}$) metallic splitrings are distributed in a domain $\Omega\subset \mathbb{R}^3$. We determine the effective behavior of this metamaterial in the limit $\eta\searrow 0$. For $\eta>0$, each single conductor occupies a simply connected domain, but the conductor closes to a ring in the limit $\eta\searrow 0$. This change of topology allows for an extra dimension in the solution space of the corresponding cellproblem. Even though both original materials (metal and void) have the same positive magnetic permeability $\mu_0>0$, we show that the effective Maxwell system exhibits, depending on the frequency, a negative magnetic response. Furthermore, we demonstrate that combining the splitring array with thin, highly conducting wires can effectively provide a negative index metamaterial. 
Claude Le Bris CERMICS Ecole Nationale des Ponts et Chaussées,
Homogenization in the presence of defects

Maria NeussRadu FriedrichAlexanderUniversität ErlangenNürnberg
Mathematical modeling and multiscale analysis of transport processes through membranes

Thomas Niedermayer Physikalisch Technische Bundesanstalt (PTB) Berlin
Macroscopic properties of a simple stochastic process with aging: Recursive solution

Daniel Peterseim Universität Augsburg
Numerical homogenization by localized orthogonal decomposition and connections to the mathematical theory of homogenization