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 $\mathbb{S}^d$ into itself in the case $d \ge 2$. This provides the answer for $d \ge 2$ to a question raised by Brezis. The problem is still open for $d=1$

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 $0$. 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

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