Afternoon workshop on Analysis and Probability

2017/08/04 Fri


14:00-14:40 Zhen-Qing Chen (University of Washington)
Markov processes with darning and their approximations

14:50-15:30 Nina Gantert (TU Munich)
Monotonicity of the speed of biased random walk among random conductances

15:40- 16:20 Stefan Neukamm (TU Dresden)
Quantitative homogenization in nonlinear elasticity for small loads

Abstract (S. Neukamm):
We consider a nonlinear elastic composite with a periodic micro-structure described by the non-convex integral functional
$$I_\varepsilon(u)=\int_\Omega W\left(\frac{x}\varepsilon,\nabla
u(x)\right)-f(x)\cdot u(x)\,dx,$$
where $u:\Omega\to\mathbb R^d$ is the deformation, $f:\Omega\to\mathbb R^d$ is an external force, $\varepsilon>0$ denotes the size of the micro-structure, and $W(y,F)$ is a stored energy function which is periodic in $y$. As it is well-known, under suitable growth conditions, $I_\varepsilon$ $\Gamma$-converges to a functional with a homogenized energy density $W_{{\sf hom}}(F)$, which is given by an \textit{infinite-cell formula}. Under appropriate assumptions on $W$ (namely, $p\geq d$-growth from below, frame indifference, minimality at identity, non-degeneracy and smoothness in a neighborhood close to the set of rotations) and on the microstructure, we show that in a neighbourhood of rotations the homogenized stored energy function $W_{{\sf hom}}$ is of class $C^2$ and characterized by a \textit{single-cell homogenization formula}. Moreover, for small data, we establish an estimate on the homogenization error, which measures the distance between (almost) minimizers $u_\varepsilon$ of $I_\varepsilon$ and the minimizer of the homogenized problem. More precisely, we prove that the $L^2$-error as well as the $H^1$-error of the associated two-scale expansion decays with the rate $\sqrt\varepsilon$.

Ryoki Fukushima (Kyoto University)
Takashi Kumagai (Kyoto University)