We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in R n , n ∈ {1, 2, 3}. Standard homogenization theory provides, for the limit of a small periodicity length ε > 0, an effective second order wave equation that describes solutions on time intervals [0, T ]. In order to approximate solutions on large time intervals [0, T ε −2 ], one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation and an estimate for errors between the solution of the original heterogeneous problem and the solution of the dispersive wave equation. We use Bloch-wave analysis to identify a family of relevant limit models and introduce an approach to select a well-posed effective model under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests.

We study the long time behavior of waves in a strongly heterogeneous medium, starting from the one-dimensional scalar wave equation with variable coefficients. We assume that the coefficients are periodic with period ε and ε > 0 is a small length parameter. Our main result is the rigorous derivation of two different dispersive models. The first is a fourth-order equation with constant coefficients including powers of ε. In the second model, the ε-dependence is completely avoided by considering a third-order linearized Korteweg-de-Vries equation. Our result is that both simplified models describe the long time behavior well. An essential tool in our analysis is an adaption operator which modifies smooth functions according to the periodic structure of the medium.

In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. Mathematics Subject Classification

We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix a ε that is periodic with characteristic length scale ε; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions u ε well by a non-dispersive wave equation on fixed time intervals (0, T ). Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive equation with homogeneous coefficients such that its solutions w ε describe u ε well on time intervals (0, T ε −2 ). More precisely, we provide a norm and uniform error estimates of the form u ε (t)−w ε (t) ≤ Cε for t ∈ (0, T ε −2 ). They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an ε-independent equation of third order that describes dispersion along rays and we present numerical examples.

We derive the homogenization limit for time harmonic Maxwell's equations in a periodic geometry with periodicity length η > 0. The considered meta-material has a singular sub-structure: the permittivity coefficient in the inclusions scales like η −2 and a part of the substructure (corresponding to wires in the related experiments) occupies only a volume fraction of order η 2 ; the fact that the wires are connected across the periodicity cells leads to contributions in the effective system. In the limit η → 0, we obtain a standard Maxwell system with a frequency dependent effective permeability µ eff (ω) and a frequency independent effective permittivity ε eff . Our formulas for these coefficients show that both coefficients can have a negative real part, the meta-material can act like a negative index material. The magnetic activity µ eff = 1 is obtained through dielectric resonances as in previous publications. The wires are thin enough to be magnetically invisible, but, due to their connectedness property, they contribute to the effective permittivity. This contribution can be negative due to a negative permittivity in the wires.

In molecular dynamics and sampling of high dimensional Gibbs measures coarse-graining is an important technique to reduce the dimensionality of the problem. We will study and quantify the coarse-graining error between the coarse-grained dynamics and an effective dynamics. The effective dynamics is a Markov process on the coarse-grained state space obtained by a closure procedure from the coarse-grained coefficients. We obtain error estimates both in relative entropy and Wasserstein distance, for both Langevin and overdamped Langevin dynamics. The approach allows for vectorial coarse-graining maps. Hereby, the quality of the chosen coarse-graining is measured by certain functional inequalities encoding the scale separation of the Gibbs measure. The method is based on error estimates between solutions of (kinetic) Fokker-Planck equations in terms of large-deviation rate functionals.Here X t ∈ d is the state of the system at time t, V is a potential, β = 1/(k B T a ) is the inverse temperature, W d t is a d-dimensional Brownian motion and X 0 is the initial state of the system.

We analyze the time harmonic Maxwell's equations in a complex geometry. The homogenization process is performed in the case that many small, thin conductors are distributed in a subdomain of R 3 . Each single conductor is, topologically, a split ring resonator, but we allow arbitrary flat shapes. In the limit of large conductivities in the rings and small ring diameters we obtain an effective Maxwell system. Depending on the frequency, the effective system can exhibit a negative effective permeability.

We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on Z d , d > 2. The model is obtained from the classical {0, 1}-Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result in [GO11], where uniformly elliptic conductances are treated, to the degenerate case. With regard to the associated random conductance model, we obtain as a side result that the corrector not only grows sublinearly, but slower than any polynomial rate. Our argument combines a quantification of ergodicity by means of a Spectral Gap on Glauber dynamics with regularity estimates on the gradient of the elliptic Green's function.