arXiv subjects
Publications and source records attributed to Hoai-Minh Nguyen.
In this paper, we present the proof of superlensing an arbitrary object using complementary media and we study reflecting complementary media for electromagnetic waves. The analysis is based on the reflecting technique and new results on the compactness, existence, and stability for the Maxwell equations with low regularity data.
In this paper, we investigate the limiting absorption principle associated to and the well-posedness of the Helmholtz equations with sign changing coefficients which are used to model negative index materials. Using the reflecting technique introduced in \cite{Ng-Complementary}, we first derive Cauchy problems from these equations. The limiting absorption principle and the well-posedness are then obtained via various a priori estimates for these Cauchy problems. There approaches are proposed to obtain the a priori estimates. The first one follows from a priori estimates of elliptic systems equipped general complementing boundary conditions due to Agmon, Douglis, and Nirenberg in their classic work \cite{ADNII}. The second approach, which is complement to the first one, is variational and based on the Dirichlet principle. The last approach, which is complement to the second one, is also variational and uses the multiplier technique. As a consequence, we obtain new results on the well-posedness of these equations for which the positivity conditions concerning the matrix-valued coefficients are imposed partially or not strictly on the interface of sign changing coefficients. As a result, the well-posedness can hold even in the case the contrast of the coefficients across the sign changing interfaces is arbitrary. This allows us to rediscover and extend largely known results obtained by the integral method, the pseudo differential operator theory, and the T-coercivity approach. The unique solution, obtained by the limiting absorption principle, is {\bf not} in $H^1_{\loc}(\mR^d)$ as usual and possibly {\bf not even} in $L^2_{\loc}(\mR^d)$. The optimality of our results is also discussed.
Cloaking a source via anomalous localized resonance (ALR) was discovered by Milton and Nicorovici in \cite{MiltonNicorovici}. A general setting in which cloaking a source via ALR takes place is the settting of doubly complementary media. This was introduced and studied in \cite{Ng-CALR} for the quasistatic regime. In this paper, we study cloaking a source via ALR for doubly complementary media in the finite frequency regime as a natural continuation of \cite{Ng-CALR}. We establish the following results: 1) Cloaking a source via ALR appears if and only if the power blows up; 2) The power blows up if the source is ``placed" near the plasmonic structure; 3) The power remains bounded if the source is far away from the plasmonic structure. Concerning the analysis, we extend ideas from \cite{Ng-CALR} and add new insights on the problem which allows us to overcome difficulties related to the finite frequency regime and to obtain new information on the problem. In particular, we are able to characterize the behaviour of the fields far enough from the plasmonic shell as the loss goes to 0 for an {\bf arbitrary source} outside the core-shell structure in the doubly complementary media setting.
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.
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.
This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two and three dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovicci in [21] and attracted a lot attention in the scientific community. Two key figures of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance and the blow up of the power of the fields as the loss goes to 0. An important class of negative index materials for which the localized resonance might appear is the class of reflecting complementary media introduced in [24]. It was showed in [29] that complementary property of media is not enough to ensure a connection between the blow up of the power and the localized resonance. In this paper, we study CALR for a subclass of complementary media called the class of doubly complementary media. This class is rich enough to allow us to cloak an arbitrary source concentrating on an arbitrary smooth bounded manifold of codimension 1 placed in an arbitrary medium via anomalous localized resonance. The following three properties are established for doubly complementary media: 1) CALR appears if and only if the power blows up; 2) The power blows up if the source is near the plasmonic structure; 3) The power remains bounded if the source is far away from the plasmonic structure. Property 2), the blow up of the power, is in fact established for reflecting complementary media...
Cloaking using complementary media was suggested by Lai et al. This was proved in the quasistatic regime by H. M. Nguyen. One of the difficulties in the study of this problem is the appearance of the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others as the loss goes to 0. To this end, the author introduced the technique of removing localized singularity and used a standard three spheres inequality. This method also works for the Helmholtz equation. However, it requires small size of the cloaked region for large frequency due to the use of the (standard) three spheres inequality. In this paper, we give a proof of cloaking using complementary media in the finite frequency regime without imposing any condition on the cloaked region; hence the cloak works for all frequency. To successfully apply the approach of H.M. Nguyen, we establish a new three spheres inequality. A modification of the cloaking setting to obtain illusion optics is also discussed.
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.
This paper is devoted to the problem of recovering a potential $q$ in a domain in $\mathbb{R}^d$ for $d \geq 3$ from the Dirichlet to Neumann map. This problem is related to the inverse Calderón conductivity problem via the Liouville transformation. It is known from the work of Haberman and Tataru [11] and Nachman and Lavine [17] that uniqueness holds for the class of conductivities of one derivative and the class of $W^{2,d/2}$ conductivities respectively. The proof of Haberman and Tataru is based on the construction of complex geometrical optics (CGO) solutions initially suggested by Sylvester and Uhlmann [22], in functional spaces introduced by Bourgain [2]. The proof of the second result, in the work of Ferreira et al. [10], is based on the construction of CGO solutions via Carleman estimates. The main goal of the paper is to understand whether or not an approach which is based on the construction of CGO solutions in the spirit of Sylvester and Uhlmann and involves only standard Sobolev spaces can be used to obtain these results. In fact, we are able to obtain a new proof of uniqueness for the Calderón problem for 1) a slightly different class as the one in [11], and for 2) the class of $W^{2,d/2}$ conductivities. The proof of statement 1) is based on a new estimate for CGO solutions and some averaging estimates in the same spirit as in [11]. The proof of statement 2) is on the one hand based on a generalized Sobolev inequality due to Kenig et al. [14] and on another hand, only involves standard estimates for CGO solutions [22]. We are also able to prove the uniqueness of a potential for 3) the class of $W^{s, 3/s}$ ($\supsetneqq W^{2, 3/2}$) conductivities with $3/2 < s < 2$ in three dimensions. As far as we know, statement 3) is new.
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 Hörmander 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.
We analyse dissipative boundary conditions for nonlinear hyperbolic systems in one space dimension. We show that a previous known sufficient condition for exponential stability with respect to the C^1-norm is optimal. In particular a known weaker sufficient condition for exponential stability with respect to the H^2-norm is not sufficient for the exponential stability with respect to the C^1-norm. Hence, due to the nonlinearity, even in the case of classical solutions, the exponential stability depends strongly on the norm considered. We also give a new sufficient condition for the exponential stability with respect to the W^{2,p}-norm. The methods used are inspired from the theory of the linear time-delay systems and incorporate the characteristic method.
This paper is devoted to the study of the behavior of the unique solution $u_δ\in H^{1}_{0}(Ω)$, as $δ\to 0$, to the equation \begin{equation*} \dive(\epss_δA \nabla u_δ) + k^2 \epss_0 Σu_δ = \epss_0 f \mbox{in} Ω, \end{equation*} where $Ω$ is a smooth connected bounded open subset of $\mR^d$ with $d=2$ or 3, $f \in L^2(Ω)$, $k$ is a non-negative constant, $A$ is a uniformly elliptic matrix-valued function, $Σ$ is a real function bounded above and below by positive constants, and $\epss_δ$ is a complex function whose {\bf the real part takes the value 1 and -1}, and the imaginary part is positive and converges to 0 as $δ$ goes to 0. This is motivated from a result in \cite{NicoroviciMcPhedranMilton94} and the concept of complementary suggested in \cite{LaiChenZhangChanComplementary, PendryNegative, PendryRamakrishna}. After introducing the reflecting complementary media, complementary media generated by reflections, we characterize $f$ for which $\|u_δ\|_{H^1(Ω)}$ remains bounded as $δ$ goes to 0. For such an $f$, we also show that $u_δ$ converges weakly in $H^1(Ω)$ and provide a formula to compute the limit.
In this paper, we study approximate cloaking for the Helmholtz equation in 3d where the cloaking device is based on transformations which blow up a cylinder of fixed height and small cross section of radius $\eps$ into the cloaked region. Assuming the zero Dirichlet boundary condition is imposed on the boundary of the cloaked region, we show that the degree of visibility is of order $\eps$ as $\eps$ goes to 0. This fact is quite surprising since it is known that the degree of visibility, for the scheme using transformations which blow up a small region of diameter $\eps$ into the cloaked region, is of order $\eps$ in 3d and $1/ |\ln \eps|$ in 2d. To understand the relation between these contexts, we as well revisit the known estimates and show that the degree of visibility is of order $\eps^{d-1}$ ($d=2,3$) for the scheme using transformations which blow up a small {\it ball} of diameter $\eps$ into the cloaked region as long as the zero Dirichlet boundary condition is imposed on the boundary of the cloaked region. The symmetry of the cross section and of the ball are crucial so that these results hold.
In this paper, we study approximate cloaking of active devices for the Helmholtz equation in the whole space of dimension 2 or 3 using the scheme introduced by Kohn, Shen, Vogelius, and Weinstein in \cite{KohnShenVogeliusWeinstein}. More precisely, we assess the degree of invisibility, determine the limit of the field inside the cloaked and cloaking regions, and show that the scheme is unstable with respect to the material parameters. As a consequence, we obtain some feasible properties of "perfect" cloaking.