1,774 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
Generalized Impedance Boundary Conditions for Strongly Absorbing Obstacles: the full Wave Equations
This paper is devoted to the study of the generalized impedance boundary
conditions (GIBCs) for a strongly absorbing obstacle in the {\bf time} regime
in two and three dimensions. The GIBCs in the time domain are heuristically
derived from the corresponding conditions in the time harmonic regime. The
latters are frequency dependent except the one of order 0; hence the formers
are non-local in time in general. The error estimates in the time regime can be
derived from the ones in the time harmonic regime when the frequency dependence
is well-controlled. This idea is originally due to Nguyen and Vogelius in
\cite{NguyenVogelius2} for the cloaking context. In this paper, we present the
analysis to the GIBCs of orders 0 and 1. To implement the ideas in
\cite{NguyenVogelius2}, we revise and extend the work of Haddar, Joly, and
Nguyen in \cite{HJNg1}, where the GIBCs were investigated for a fixed frequency
in three dimensions. Even though we heavily follow the strategy in
\cite{NguyenVogelius2}, our analysis on the stability contains new ingredients
and ideas. First, instead of considering the difference between solutions of
the exact model and the approximate model, we consider the difference between
their derivatives in time. This simple idea helps us to avoid the machinery
used in \cite{NguyenVogelius2} concerning the integrability with respect to
frequency in the low frequency regime. Second, in the high frequency regime,
the Morawetz multiplier technique used in \cite{NguyenVogelius2} does not fit
directly in our setting. Our proof makes use of a result by H\"ormander in
\cite{Hor}. Another important part of the analysis in this paper is the
well-posedness in the time domain for the approximate problems imposed with
GIBCs on the boundary of the obstacle, which are non-local in time
Localized and complete resonance in plasmonic structures
This paper studies a possible connection between the way the time averaged
electromagnetic power dissipated into heat blows up and the anomalous localized
resonance in plasmonic structures. We show that there is a setting in which the
localized resonance takes place whenever the resonance does and moreover, the
power is always bounded and might go to . We also provide another setting in
which the resonance is complete and the power goes to infinity whenever
resonance occurs; as a consequence of this fact there is no localized
resonance. This work is motivated from recent works on cloaking via anomalous
localized resonance
Discreteness of interior transmission eigenvalues revisited
This paper is devoted to the discreteness of the transmission eigenvalue
problems. It is known that this problem is not self-adjoint and a priori
estimates are non-standard and do not hold in general. Two approaches are used.
The first one is based on the multiplier technique and the second one is based
on the Fourier analysis. The key point of the analysis is to establish the
compactness and the uniqueness for Cauchy problems under various conditions.
Using these approaches, we are able to rediscover quite a few known
discreteness results in the literature and obtain various new results for which
only the information near the boundary are required and there might be no
contrast of the coefficients on the boundary
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