172,514 research outputs found
Homogenization of resonant chiral metamaterials
Homogenization of metamaterials is a crucial issue as it allows to describe
their optical response in terms of effective wave parameters as e.g.
propagation constants. In this paper we consider the possible homogenization of
chiral metamaterials. We show that for meta-atoms of a certain size a critical
density exists above which increasing coupling between neighboring meta-atoms
prevails a reasonable homogenization. On the contrary, a dilution in excess
will induce features reminiscent to photonic crystals likewise prevailing a
homogenization. Based on Bloch mode dispersion we introduce an analytical
criterion for performing the homogenization and a tool to predict the
homogenization limit. We show that strong coupling between meta-atoms of chiral
metamaterials may prevent their homogenization at all.Comment: 8 pages, 7 figure
An introduction to the qualitative and quantitative theory of homogenization
We present an introduction to periodic and stochastic homogenization of
ellip- tic partial differential equations. The first part is concerned with the
qualitative theory, which we present for equations with periodic and random
coefficients in a unified approach based on Tartar's method of oscillating test
functions. In partic- ular, we present a self-contained and elementary argument
for the construction of the sublinear corrector of stochastic homogenization.
(The argument also applies to elliptic systems and in particular to linear
elasticity). In the second part we briefly discuss the representation of the
homogenization error by means of a two- scale expansion. In the last part we
discuss some results of quantitative stochastic homogenization in a discrete
setting. In particular, we discuss the quantification of ergodicity via
concentration inequalities, and we illustrate that the latter in combi- nation
with elliptic regularity theory leads to a quantification of the growth of the
sublinear corrector and the homogenization error.Comment: Lecture notes of a minicourse given by the author during the GSIS
International Winter School 2017 on "Stochastic Homogenization and its
applications" at the Tohoku University, Sendai, Japan; This version contains
a correction of Lemma 2.1
Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations
We present exponential error estimates and demonstrate an algebraic
convergence rate for the homogenization of level-set convex Hamilton-Jacobi
equations in i.i.d. random environments, the first quantitative homogenization
results for these equations in the stochastic setting. By taking advantage of a
connection between the metric approach to homogenization and the theory of
first-passage percolation, we obtain estimates on the fluctuations of the
solutions to the approximate cell problem in the ballistic regime (away from
flat spot of the effective Hamiltonian). In the sub-ballistic regime (on the
flat spot), we show that the fluctuations are governed by an entirely different
mechanism and the homogenization may proceed, without further assumptions, at
an arbitrarily slow rate. We identify a necessary and sufficient condition on
the law of the Hamiltonian for an algebraic rate of convergence to hold in the
sub-ballistic regime and show, under this hypothesis, that the two rates may be
merged to yield comprehensive error estimates and an algebraic rate of
convergence for homogenization.
Our methods are novel and quite different from the techniques employed in the
periodic setting, although we benefit from previous works in both first-passage
percolation and homogenization. The link between the rate of homogenization and
the flat spot of the effective Hamiltonian, which is related to the
nonexistence of correctors, is a purely random phenomenon observed here for the
first time.Comment: 57 pages. Revised version. To appear in J. Amer. Math. So
Stochastic homogenization of nonconvex unbounded integral functionals with convex growth
We consider the well-travelled problem of homogenization of random integral
functionals. When the integrand has standard growth conditions, the qualitative
theory is well-understood. When it comes to unbounded functionals, that is,
when the domain of the integrand is not the whole space and may depend on the
space-variable, there is no satisfactory theory. In this contribution we
develop a complete qualitative stochastic homogenization theory for nonconvex
unbounded functionals with convex growth. We first prove that if the integrand
is convex and has -growth from below (with , the dimension), then it
admits homogenization regardless of growth conditions from above. This result,
that crucially relies on the existence and sublinearity at infinity of
correctors, is also new in the periodic case. In the case of nonconvex
integrands, we prove that a similar homogenization result holds provided the
nonconvex integrand admits a two-sided estimate by a convex integrand (the
domain of which may depend on the space-variable) that itself admits
homogenization. This result is of interest to the rigorous derivation of rubber
elasticity from polymer physics, which involves the stochastic homogenization
of such unbounded functionals.Comment: 64 pages, 2 figure
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