3,322 research outputs found
Superlensing using complementary media
This paper studies magnifying superlens using complementary media.
Superlensing using complementary media was suggested by Veselago in [16] and
innovated by Nicorovici et al. in [9] and Pendry in [10]. The study of this
problem is difficult due to two facts. Firstly, this problem is unstable since
the equations describing the phenomena have sign changing coefficients; hence
the ellipticity is lost. Secondly, the phenomena associated are localized
resonant, i.e., the field explodes in some regions and remains bounded in some
others. This makes the problem difficult to analyse. In this paper, we develop
the technique of removing of localized singularity introduced in [6] and make
use of the reflecting technique in [5] to overcome these two difficulties. More
precisely, we suggest a class of lenses which has root from [9] and [14] and
inspired from [6] and give a proof of superlensing for this class. To our
knowledge, this is the first rigorous proof on the magnification of an
arbitrary inhomogeneous object using complementary media.Comment: Appeared in AIH
Cloaking using complementary media in the quasistatic regime
Cloaking using complementary media was suggested by Lai et al. in [8]. The
study of this problem faces two difficulties. Firstly, this problem is unstable
since the equations describing the phenomenon have sign changing coefficients,
hence the ellipticity is lost. Secondly, the localized resonance, i.e., the
field explodes in some regions and remains bounded in some others, might
appear. In this paper, we give a proof of cloaking using complementary media
for a class of schemes inspired from [8] in the quasistatic regime. To handle
the localized resonance, we introduce the technique of removing localized
singularity and apply a three spheres inequality. The proof also uses the
reflecting technique in [11]. To our knowledge, this work presents the first
proof on cloaking using complementary media.Comment: To appear in AIH
A refined estimate for the topological degree
We sharpen an estimate of Bourgain, Brezis, and Nguyen for the topological
degree of continuous maps from a sphere into itself in the case
. This provides the answer for to a question raised by
Brezis. The problem is still open for
On anisotropic Sobolev spaces
We investigate two types of characterizations for anisotropic Sobolev and BV
spaces. In particular, we establish anisotropic versions of the
Bourgain-Brezis-Mironescu formula, including the magnetic case both for Sobolev
and BV functions.Comment: 10 page
Some remarks on rearrangement for nonlocal functionals
We prove that a nonlocal functional approximating the standard Dirichlet
-norm fails to decrease under two-point rearrangement. Furthermore, we get
other properties related to this functional such as decay and compactness, and
the Polya-Szeg\"o inequality for Riesz fractional gradients, a notion recently
introduced in the literature.Comment: 12 page
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