Abstract:
Scaling laws for characteristic length scales (in time or in the model parameters) are both experimentally robust and accessible for rigorous analysis. In multiscale situations cross--overs between different scaling laws are observed. We give a simple example from micromagnetics. In soft ferromagnetic films, the geometric character of a wall separating two magnetic domains depends on the film thickness. We identify this transition from a N\'eel wall to an Asymmetric Bloch wall by rigorously establishing a cross--over in the specific wall energy.

Abstract:
We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters $\eps$ and $\eta$ and defined over $S^2-$vector fields $m$ that are tangent at the boundary of a two-dimensional domain $\Omega$. We are interested in the behavior of minimizers as $\eps, \eta \to 0$. The minimizers tend to be in-plane away from a region of length scale $\eps$ (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that $S^1-$transition layers of length scale $\eta$ (N\'eel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of N\'eel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields $m_{\eps, \eta}$ of energies close to the Landau state in the regime where a vortex is energetically more expensive than a N\'eel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of $S^2-$vector fields by $S^1-$vector fields away from the vortex balls.

Abstract:
We consider a random, uniformly elliptic coefficient field $a(x)$ on the $d$-dimensional integer lattice $\mathbb{Z}^d$. We are interested in the spatial decay of the quenched elliptic Green function $G(a;x,y)$. Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble $\langle\cdot\rangle$. We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, $\langle|\nabla_x G(x,y)|^p\rangle$ and $\langle|\nabla_x\nabla_y G(x,y)|^p\rangle$, have the same decay rates in $|x-y|\gg 1$ as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel \cite{DeuschelDelmotte}, which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of $G$, that is, $\langle|\nabla_x G(x,y)|^2\rangle$ and $\langle|\nabla_x\nabla_y G(x,y)|\rangle$. As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.

Abstract:
We consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of a finite range of dependence. We show that the gradient $\nabla\phi$ of the corrector $\phi$, when spatially averaged over a scale $R\gg 1$ decays like $R^{-\alpha}$ for any $\alpha<\frac{d}{2}$. We establish these rates on the level of Gaussian bounds in terms of the stochastic integrability.

Abstract:
We prove some interpolation inequalities which arise in the analysis of pattern formation in physics. They are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts (coarsening and branching in micromagnetics and superconductors). The main ingredient in the proof of our inequalities is a geometric construction which was first used by Choksi, Conti, Kohn, and one of the authors in the study of branching in superconductors.

Abstract:
We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb{Z}^d$. Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions $d>2$ and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages - the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions.

Abstract:
We consider an elliptic system of equations on the torus $\left[ -\frac{L}{2}, \frac{L}{2} \right)^d$ with random coefficients $A$, that are assumed to be coercive and stationary. Using two different approaches we obtain moment bounds on the gradient of the corrector, independent of the domain size $L$. In the first approach we use Green function representation. For that we require $A$ to be locally H\"older continuous and distribution of $A$ to satisfy Logarithmic Sobolev inequality. The second method works for non-smooth (possibly discontinuous) coefficients, and it requires that statistics of $A$ satisfies Spectral Gap estimate.

Abstract:
In the present contribution we establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergence form, when the diffusion coefficients satisfy a spectral gap estimate in probability, and for $d>2$. The main difference with respect to the first part of [Gloria-Otto, arXiv:1409.0801] is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory.

Abstract:
We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2\pi(u)=\xi$ driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that $\pi'$ is bounded away from zero and infinity. Under the further assumption of Lipschitz continuity of $\pi'$ we show that the stationary solution is - as for the linear case - almost surely H\"older continuous with exponent $\alpha$ for any $\alpha<\frac{1}{2}$ w. r. t. the parabolic metric. More precisely, we show that the corresponding local H\"older norm has stretched exponential moments. On the stochastic side, we use a combination of martingale arguments to get second moment estimates with concentration of measure arguments to upgrade to Gaussian moments. On the deterministic side, we first perform a Campanato iteration based on the De Giorgi-Nash Theorem as well as finite and infinitesimal versions of the $H^{-1}$-contraction principle, which yields Gaussian moments for a weaker H\"older norm. In a second step this estimate is improved to the optimal H\"older exponent at the expense of weakening the integrability to stretched exponential.

Abstract:
This paper is the companion article to [Ann. Probab. 39 (2011) 779--856]. We consider a discrete elliptic equation on the $d$-dimensional lattice $\mathbb{Z}^d$ with random coefficients $A$ of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric "homogenized" matrix $A_{\mathrm{hom}}=a_{\mathrm{hom}}\mathrm{Id}$ is characterized by $\xi\cdot A_{\mathrm{hom}}\xi=<(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)>$ for any direction $\xi\in\mathbb{R}^d$, where the random field $\phi$ (the "corrector") is the unique solution of $-\nabla^*\cdot A(\xi+\nabla\phi)=0$ in $\mathbb{Z}^d$ such that $\phi(0)=0$, $\nabla\phi$ is stationary and $<\nabla\phi>=0$, $<\cdot>$ denoting the ensemble average (or expectation).