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In this paper, we establish the discreteness of transmission eigenvalues for Maxwell's equations. More precisely, we show that the spectrum of the transmission eigenvalue problem is discrete, if the electromagnetic parameters $\eps, \, \mu, \, \heps, \, \hmu$ in the equations characterizing the inhomogeneity and background, are smooth in some neighborhood of the boundary, isotropic on the boundary, and satisfy the conditions $\eps \neq \heps$, $\mu \neq \hmu$, and $\eps/ \mu \neq \heps/ \hmu$ on the boundary. These are quite general assumptions on the coefficients which are easy to check. To our knowledge, our paper is the first to establish discreteness of transmission eigenvalues for Maxwell's equations without assuming any restrictions on the sign combination of the contrasts $\eps-\heps$ and $\mu - \hmu$ near the boundary, and allowing for all the electromagnetic parameters to be inhomogeneous and anisotropic, except for on the boundary where they are isotropic but not necessarily constant as it is often assumed in the literature.

Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreteness of the set of eigenvalues of the transmission problem and studied their locations. In this paper, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions, assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert-Schmidt operators.

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.

We study the approximate cloaking via transformation optics for electromagnetic waves in the time harmonic regime in which the cloaking device 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 analyze 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 achieved in both cases. Moreover, the degree of visibility 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.

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.

This is a survey on approximate cloaking using transformation optics on acoustic and electromagnetic waves. Both the time-harmonic and the time regimes are discussed.

In this paper, we establish approximate cloaking for the heat equation via transformation optics. We show that the degree of visibility is of the order $\varepsilon$ in three dimensions and $|\ln \varepsilon|^{-1}$ in two dimensions, where $\varepsilon$ is the regularization parameter. To this end, we first transform the problem in time domain into a family of problems in frequency domain by taking the Fourier transform with respect to time, and then derive appropriate estimates in the frequency domain.

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.

International audience; This paper is devoted to the controllability of a general linear hyperbolic system in one space dimension using boundary controls on one side. Under precise and generic assumptions on the boundary conditions on the other side, we previously established the optimal time for the null and the exact controllability for this system for a generic source term. In this work, we prove the null-controllability for any time greater than the optimal time and for any source term. Similar results for the exact controllability are also discussed.

We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the $L^2$-norm if their initial data belong to a finite dimensional subspace $\M$. In this paper, we show that all solutions of the nonlinear KdV system decay to 0 at least with the rate $1/ t^{1/2}$ when $\dim \M = 1$ or when $\dim \M$ is even and a specific condition is satisfied, provided that their initial data is sufficiently small. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. As a consequence, we rediscover known results which were previously established for $\dim \M = 1$ or for the smallest critical length $L$ with $\dim \M = 2$ by a different approach using the center manifold theory, and obtain new results. We also show that the decay rate is not slower than $\ln (t + 2) / t$ for all critical lengths.

International audience; We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.

The aim of this note is to survey recent results contained in Nguyen H-M, Squassina M. [On anisotropic Sobolev spaces. Commun Contemp Math, to appear. DOI:10.1142/S0219199718500177]; Nguyen H-M, Pinamonti A, Squassina M, etal. [New characterizations of magnetic Sobolev spaces. Adv Nonlinear Anal. 2018;7(2):227–245]; Pinamonti A, Squassina M, Vecchi E. [Magnetic BV functions and the Bourgain-Brezis-Mironescu formula. Adv Calc Var, to appear. DOI:10.1515/acv-2017-0019]; Pinamonti A, Squassina M, Vecchi E. [The Maz'ya-Shaposhnikova limit in the magnetic setting. J Math Anal Appl. 2017;449:1152–1159] and Squassina M, Volzone B. [Bourgain-Brezis-Mironescu formula for magnetic operators. C R Math Acad Sci Paris. 2016;354:825–831], where the authors extended to the magnetic setting several characterizations of Sobolev and BV functions.

In this paper, we discuss our recent works on the null-controllability, the exact controllability, and the stabilization of linear hyperbolic systems in one dimensional space using boundary controls on one side for the optimal time. Under precise and generic assumptions on the boundary conditions on the other side, we first obtain the optimal time for the null and the exact controllability for these systems for a generic source term. We then prove the null-controllability and the exact controllability for any time greater than the optimal time and for any source term. Finally, for homogeneous systems, we design feedbacks which stabilize the systems and bring them to the zero state at the optimal time. Extensions for the non-linear homogeneous system are also discussed, Comment: To appear in the proceedings of the 2020 59th IEEE Conference on Decision and Control. arXiv admin note: text overlap with arXiv:1910.12268

We establish a full range of Caffarelli–Kohn–Nirenberg inequalities and their variants for fractional Sobolev spaces.

The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this problem are now known to depend on a type of contrast between coefficients near the boundary. Previously, we established the discreteness of eigenvalues for a large class of anisotropic coefficients which is related to the celebrated complementing conditions due to Agmon, Douglis, and Nirenberg. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for this class of coefficients under an additional mild assumption on the continuity of the coefficients. The analysis is new and based on the $L^p$ regularity theory for the transmission problem established here. It also involves a subtle application of the spectral theory for the Hilbert Schmidt operators. Our work extends largely known results in the literature which are mainly devoted to the isotropic case with $C^\infty$-coefficients., Comment: 26 pages

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., Comment: 21 pages

In this paper, we present various schemes of cloaking an arbitrary object 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 the two dimensional quasi-static regime an annular plasmonic structure of coefficient $-1$ cloaks small but finite size objects nearby. 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.

We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field $A$ is differentiable and its exterior derivative corresponding to the magnetic field $dA$ is bounded. In particular, we prove that, for $d \ge 1$ and $p>1$, the trace of the magnetic Sobolev space $W^{1, p}_A(\mathbb{R}^{d+1}_+)$ is exactly $W^{1-1/p, p}_{A^{\shortparallel}}(\mathbb{R}^d)$ where $A^{\shortparallel}(x) =( A_1, \dotsc, A_d)(x, 0)$ for $x \in \mathbb{R}^d$ with the convention $A = (A_1, \dotsc, A_{d+1})$ when $A \in C^1(\overline{\mathbb{R}^{d+1}_+}, \mathbb{R}^{d+1})$. We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a half-space to the entire space., 24 pages

We study the limiting absorption principle and the well-posedness of Maxwell equations with anisotropic sign-changing coefficients in the time-harmonic domain. The starting point of the analysis is to obtain Cauchy problems associated with two Maxwell systems using a change of variables. We then derive a priori estimates for these Cauchy problems using two different approaches. The Fourier approach involves the complementing conditions for the Cauchy problems associated with two elliptic equations, which were studied in a general setting by Agmon, Douglis, and Nirenberg. The variational approach explores the variational structure of the Cauchy problems of the Maxwell equations. As a result, we obtain general conditions on the coefficients for which the limiting absorption principle and the well-posedness hold. Moreover, these {\it new} conditions are of a local character and easy to check. Our work is motivated by and provides general sufficient criteria for the stability of electromagnetic fields in the context of negative-index metamaterials.

We investigate cloaking property of negative-index metamaterials in the time-harmonic electromagnetic setting for the so-called doubly complementary media. These are media consisting of negative-index metamaterials in a shell (plasmonic structure) and positive-index materials in its complement for which the shell is complementary to a part of the core and a part of the exterior of the core-shell structure. We show that an arbitrary object is invisible when it is placed close to a plasmonic structure of a doubly complementary medium as long as its cross section is smaller than a threshold given by the property of the plasmonic structure. To handle the loss of the compactness and of the ellipticity of the modeling Maxwell equations with sign-changing coefficients, we first obtain Cauchy's problems associated with two Maxwell systems using reflections. We then derive information from them, and combine it with the removing localized singularity technique to deal with the localized resonance. A central part of the analysis on the Cauchy's problems is to establish three-sphere inequalities with partial data for general elliptic systems, which are interesting in themselves. The proof of these inequalities first relies on an appropriate change of variables, inspired by conformal maps, and is then based on Carleman's estimates for a class of degenerate elliptic systems., Comment: Heuristic arguments and some figures are added

This paper concerns approximate cloaking by mapping for a full, but scalar wave equation, when one allows for physically relevant frequency dependence of the material properties of the cloak. The paper is a natural continuation of [20] , but here we employ the Drude–Lorentz model in the cloaking layer, that is otherwise constructed by an approximate blow up transformation of the type introduced in [10] . The central mathematical problem is to analyze the effect of a small inhomogeneity in the context of this non-local full wave equation.

Inspired by the BBM formula and by work of G. Leoni and D. Spector, we analyze the asymptotic behavior of two sequences of convex nonlocal functionals ( Ψ n ( u ) ) and ( Φ n ( u ) ) which converge formally to the BV-norm of u . We show that pointwise convergence when u is not smooth can be delicate; by contrast, Γ -convergence to the BV-norm is a robust and very useful mode of convergence.

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

We study the [Formula: see text]-convergence of a family of non-local, non-convex functionals in [Formula: see text] for [Formula: see text], where [Formula: see text] is an open interval. We show that the limit is a multiple of the [Formula: see text] semi-norm to the power [Formula: see text] when [Formula: see text] (respectively, the [Formula: see text] semi-norm when [Formula: see text]). In dimension one, this extends earlier results which required a monotonicity condition.

We are concerned about the controllability of a general linear hyperbolic system of the form $\partial_t w (t, x) = \Sigma(x) \partial_x w (t, x) + \gamma C(x) w(t, x) $ ($\gamma \in \mR$) 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 $\gamma$. We also present examples which yield that the generic requirement is necessary. In the case of constant $\Sigma$ and of two positive directions, we prove that the null-controllability is attained for any time greater than the optimal time for all $\gamma \in \mR$ 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.

In this paper, we establish approximate cloaking for the heat equation via transformation optics. We show that the degree of visibility is of the order ε in three dimensions and |lnε|−1 in two dimensions, where ε is the regularization parameter. To this end, we first transform the problem in time domain into a family of problems in frequency domain by taking the Fourier transform with respect to time, and then derive appropriate estimates in the frequency domain.

This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the 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 latter is 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 [Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] for the cloaking context. In this paper, we present the analysis to the GIBCs of orders 0 and 1. To implement the ideas in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], we revise and extend the work of Haddar, Joly, and Nguyen, [Generalized impedance boundary condition for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci.15 (2005) 1273–1300], where the GIBCs were investigated for a fixed frequency in three dimensions. Even though we heavily follow the strategy in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], 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 [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] concerning the integrability with respect to frequency in the low frequency regime. Second, in the high frequency regime, the Morawetz multiplier technique used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variable, SIAM J. Math. Anal.44 (2012) 769–807] does not fit directly in our setting. Our proof makes use of a result by Hörmander in [Lp estimates for (pluri-) subharmonic functions, Math. Scand.20 (1967) 65–78]. 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.

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 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 might be localized resonant, i.e., the field explodes in some regions and remains bounded in some others. This makes the problem difficult to analyze. 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. (C) 2014 Elsevier Masson SAS. All rights reserved.

This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in two and three dimensions in the quasistatic regime. Two key figures of CALR are (i) the localized resonance and (ii) 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 and analyzed in [8-10]. It was shown in [12] that the complementary property of media is not enough to ensure such a connection. In this paper, we introduce a subclass of complementary media called the class of doubly complementary media. This class is rich enough to allow us to do cloaking via anomalous localized resonance for an arbitrary source concentrating on an arbitrary smooth bounded manifold of codimension 1 located in an arbitrary medium. The following three properties are established: 1) CALR appears if and only if the power blows up; 2) the power blows up if the source is "located" near the plasmonic structure; 3) the power remains bounded if the source is far away from the plasmonic structure. Property P2), the blow up of the power, is in fact established for reflecting complementary media. (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

We provide a new characterization of the logarithmic Sobolev inequality., 5 pages

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.

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-Szeg\"o inequality for Riesz fractional gradients, a notion recently introduced in the literature., Comment: 12 pages

We sharpen an estimate of [4] for the topological degree of continuous maps from a sphere Sdinto itself in the case d >= 2. This provides the answer for d >= 2 to a question raised by Brezis. The problem is still open for d = 1. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

This paper is devoted to the study of the behavior of the unique solution $u_\delta \in H^{1}_{0}(\Omega)$, as $\delta \to 0$, to the equation \begin{equation*} \dive(\epss_\delta A \nabla u_{\delta}) + k^2 \epss_0 \Sigma u_{\delta} = \epss_0 f \mbox{in} \Omega, \end{equation*} where $\Omega$ is a smooth connected bounded open subset of $\mR^d$ with $d=2$ or 3, $f \in L^2(\Omega)$, $k$ is a non-negative constant, $A$ is a uniformly elliptic matrix-valued function, $\Sigma$ is a real function bounded above and below by positive constants, and $\epss_\delta$ 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 $\delta$ 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 $\|u_\delta\|_{H^1(\Omega)}$ remains bounded as $\delta$ goes to 0. For such an $f$, we also show that $u_\delta$ converges weakly in $H^1(\Omega)$ and provide a formula to compute the limit.

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.

International audience; We present new results concerning the approximation of the total variation, ∫Ω|∇u|∫Ω|∇u|, of a function u by non-local, non-convex functionals of the formView the MathML sourceΛδ(u)=∫Ω∫Ωδφ(|u(x)−u(y)|/δ)|x−y|d+1dxdy,Turn MathJax onas δ→0δ→0, where Ω is a domain in RdRd and φ:[0,+∞)→[0,+∞)φ:[0,+∞)→[0,+∞) is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate, and numerous problems remain open. The original motivation of our work comes from Image Processing.; Nous présentons des résultats nouveaux concernant l'approximation de la variation totale ∫Ω|∇u|∫Ω|∇u| d'une fonction u par des fonctionnelles non convexes et non locales de la formeView the MathML sourceΛδ(u)=∫Ω∫Ωδφ(|u(x)−u(y)|/δ)|x−y|d+1dxdy,Turn MathJax onquand δ→0δ→0, où Ω est un domaine de RdRd et φ:[0,+∞)→[0,+∞)φ:[0,+∞)→[0,+∞) est une fonction croissante vérifiant certaines hypothèses. Le mode de convergence est extrêmement délicat et de nombreux problèmes restent ouverts. La motivation provient du traitement d'images.

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.

We consider the deformation of a thin elastic film bonded to a thick compliant substrate, when the (compressive) misfit is far beyond critical. We take a variational viewpoint—focusing on the total elastic energy, i.e. the membrane and bending energy of the film plus the elastic energy of the substrate—viewing the buckling of the film as a problem of energy-driven pattern formation. We identify the scaling law of the minimum energy with respect to the physical parameters of the problem, and we prove that a herringbone pattern achieves the optimal scaling. These results complement previous numerical studies, which have shown that an optimized herringbone pattern has lower energy than a number of other patterns. Our results are different, because (i) we make the scaling law achieved by the herringbone pattern explicit, and (ii) we give an elementary, ansatz-free proof that no pattern can achieve a better law.

We study, in the context of the full wave equation, an approximate cloaking scheme that was previously considered for the Helmholtz equation [R. V. Kohn, D. Onofrei, M. S. Vogelius, and M. I. Weinstein, Comm. Pure Appl. Math., 63 (2010), pp. 973--1016, H.-M. Nguyen and M. S. Vogelius, Arch. Ration. Mech. Anal., 203 (2011), pp. 769--807]. This cloaking scheme consists in a combination of an absorbing layer with an anisotropic layer, obtained by so-called transformation optics. We give optimal bounds for the visibility that tend to zero as a certain regularization parameter approaches 0. Our bounds are based on recent estimates for the Helmholtz equation [from Nguyen and Vogelius], some low frequency improvements of these estimates, and the use of Fourier transformation in time. Read More: http://epubs.siam.org/doi/abs/10.1137/110833154