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Hoai-Minh Nguyen

Publications and source records attributed to Hoai-Minh Nguyen.

At least 55 records · Page 3Linked to original sources
arXiv2019

Approximate cloaking for time-dependent Maxwell equations via transformation optics

We study approximate cloaking using transformation optics for electromagnetic waves in the time domain. Our approach is based on estimates of the degree of visibility in the frequency domain for all frequencies in which the frequency dependence is explicit. The difficulty and the novelty analysis parts are in the low and high frequency regimes. To this end, we implement a variational technique in the low frequency domain, and multiplier and duality techniques in the high frequency domain. Our approach is inspired by the work of Nguyen and Vogelius on the wave equation.

math.AP↗
arXiv2018

Optimal time for the controllability of linear hyperbolic systems in one dimensional space

We are concerned about the controllability of a general linear hyperbolic system of the form $\partial_t w (t, x) = Σ(x) \partial_x w (t, x) + γC(x) w(t, x) $ ($γ\in \mR$) in one space dimension using boundary controls on one side. More precisely, we establish the optimal time for the null and exact controllability of the hyperbolic system for generic $γ$. We also present examples which yield that the generic requirement is necessary. In the case of constant $Σ$ and of two positive directions, we prove that the null-controllability is attained for any time greater than the optimal time for all $γ\in \mR$ and for all $C$ which is analytic if the slowest negative direction can be alerted by {\it both} positive directions. We also show that the null-controllability is attained at the optimal time by a feedback law when $C \equiv 0$. Our approach is based on the backstepping method paying a special attention on the construction of the kernel and the selection of controls.

math.OC↗
arXiv2018

Approximate cloaking for electromagnetic waves via transformation optics: cloaking vs infinite energy

We study the approximate cloaking via transformation optics for electromagnetic waves in the time harmonic regime in which the cloaking device {\it only} consists of a layer constructed by the mapping technique. Due to the fact that no-lossy layer is required, resonance might appear and the analysis is delicate. We analyse both non-resonant and resonant cases. In particular, we show that the energy can blow up inside the cloaked region in the resonant case and/whereas cloaking is {\it achieved} in {\it both} cases. Moreover, the degree of visibility {\it depends} on the compatibility of the source inside the cloaked region and the system. These facts are new and distinct from known mathematical results in the literature.

math.AP↗
arXiv2018

Negative index materials: some mathematical perspectives

Negative index materials are artificial structures whose refractive index has a negative value over some frequency range. These materials were postulated and investigated theoretically by Veselago in 1964 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow for the construction of negative index materials at scales that are interesting for applications, which has made them a very active topic of investigation. In this paper, we report various mathematical results on the properties of negative index materials and their applications. The topics discussed herein include superlensing using complementary media, cloaking using complementary media, cloaking an object via anomalous localized resonance, and the well-posedness and the finite speed propagation in media consisting of dispersive metamaterials. Some of the results have been refined and have simpler proofs than the original ones.

math-ph↗
arXiv2018

On Hardy and Caffarelli-Kohn-Nirenberg inequalities

We establish improved versions of the Hardy and Caffarelli-Kohn-Nirenberg inequalities by replacing the standard Dirichlet energy with some nonlocal nonconvex functionals which have been involved in estimates for the topological degree of continuous maps from a sphere into itself and characterizations of Sobolev spaces.

math.CA↗
arXiv2017

Electromagnetic wave propagation in media consisting of dispersive metamaterials

We establish the well-posedness, the finite speed propagation, and a regularity result for Maxwell's equations in media consisting of dispersive (frequency dependent) metamaterials. Two typical examples for such metamaterials are materials obeying Drude's and Lorentz' models. The causality and the passivity are the two main assumptions and play a crucial role in the analysis. It is worth noting that by contrast the well-posedness in the frequency domain is not ensured in general. We also provide some numerical experiments using the Drude's model to illustrate its dispersive behaviour.

math-ph↗
arXiv2017

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 $d=1$.

math.AP↗
arXiv2017

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.

math.FA↗
arXiv2017

Some remarks on rearrangement for nonlocal functionals

We prove that a nonlocal functional approximating the standard Dirichlet $p$-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-Szegö inequality for Riesz fractional gradients, a notion recently introduced in the literature.

math.FA↗
arXiv2017

New characterizations of magnetic Sobolev spaces

We establish two new characterizations of magnetic Sobolev spaces for Lipschitz magnetic fields in terms of nonlocal functionals. The first one is related to the BBM formula, due to Bourgain, Brezis, and Mironescu. The second one is related to the work of the first author on the classical Sobolev spaces. We also study the convergence almost everywhere and the convergence in $L^1$ appearing naturally in these contexts.

math.AP↗
arXiv2017

Cloaking using complementary media for electromagnetic waves

Negative index materials are artificial structures whose refractive index has negative value over some frequency range. The study of these materials has attracted a lot of attention in the scientific community not only because of their many potential interesting applications but also because of challenges in understanding their intriguing properties due to the sign-changing coefficients in equations describing their properties. In this paper, we establish cloaking using complementary media for electromagnetic waves. This confirms and extends the suggestions in two dimensions of Lai et al. for the full Maxwell equations. The analysis is based on the reflecting and removing localized singularity techniques, three-sphere inequalities, and the fact that the Maxwell equations can be reduced to a weakly coupled second order elliptic equations.

math.AP↗
arXiv2016

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.

math.AP↗
arXiv2016

Non-local functionals related to the total variation and connections with Image Processing

We present new results concerning the approximation of the total variation, $\int_Ω |\nabla u|$, of a function $u$ by non-local, non-convex functionals of the form $$ Λ_δu = \int_Ω \int_Ω \frac{δφ\big( |u(x) - u(y)|/ δ\big)}{|x - y|^{d+1}} \, dx \, dy, $$ as $δ\to 0$, where $Ω$ is a domain in $\mathrm{R}^d$ and $φ: [0, + \infty) \to [0, + \infty)$ is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi's concept of Gamma-convergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing.

math.OC↗
arXiv2016

Cloaking an arbitrary object via anomalous localized resonance: the cloak is independent of the object

In this paper, we present various schemes of cloaking an arbitrary objects via anomalous localized resonance and provide their analysis in two and three dimensions. This is a way to cloak an object using negative index materials in which the cloaking device is independent of the object. As a result, we show that in two dimensional quasistatic regime an annular plasmonic structure $-I$ cloaks small but finite size objects near by. We also discuss its connections with superlensing and cloaking using complementary media. In particular, we confirm the possibility that a lens can act like a cloak and conversely. This possibility was raised about a decade ago in the literature.

math-ph↗
arXiv2016

Superlensing using hyperbolic metamaterials: the scalar case

This paper is devoted to superlensing using hyperbolic metamaterials: the possibility to image an arbitrary object using hyperbolic metamaterials without imposing any conditions on size of the object and the wave length. To this end, two types of schemes are suggested and their analysis are given. The superlensing devices proposed are independent of the object. It is worth noting that the study of hyperbolic metamaterials is challenging due to the change type of modelling equations, elliptic in some regions, hyperbolic in some others.

math.AP↗
arXiv2016

The BBM formula revisited

In this paper, we revise the BBM formula due to J. Bourgain, H. Brezis, and P. Mironescu in [1].

math.CA↗