Abstract
In periodic homogenization problems, one
considers a sequence
(
u
η
)
η
{(u^{\eta})_{\eta}}
of solutions to periodic
problems and derives a homogenized equation for an effective
quantity
u
^
{\hat{u}}
. In many applications,
u
^
{\hat{u}}
is the weak
limit of
(
u
η
)
η
{(u^{\eta})_{\eta}}
, but in some applications
u
^
{\hat{u}}
must
be defined differently. In the homogenization of Maxwell’s
equations in periodic media, the effective magnetic field is
given by the geometric average of the two-scale limit. The notion
of a geometric average has been introduced in [G. Bouchitté, C. Bourel and D. Felbacq,
Homogenization of the 3D Maxwell system near resonances and artificial magnetism,
C. R. Math. Acad. Sci. Paris 347 2009, 9–10, 571–576]; it associates to a
curl-free field
Y
∖
Σ
¯
→
ℝ
3
{Y\setminus\overline{\Sigma}\to\mathbb{R}^{3}}
, where
Y is the periodicity cell and Σ an inclusion, a vector
in
ℝ
3
{\mathbb{R}^{3}}
. In this article, we extend previous definitions to
more general inclusions, in particular inclusions that are not
compactly supported in the periodicity cell. The physical
relevance of the geometric average is demonstrated by various
results, e.g., a continuity property of limits of tangential traces.
Estimation of the Bid-Ask Prices for the European Discrete Geometric Average and Arithmetic Average Asian Options