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.
We study the invisibility via anomalous localized resonance of a general source for electromagnetic waves in the setting of doubly complementary media. As a result, we show that cloaking is achieved if the power is blown up. We also reveal a critical
length for the invisibility of a source that occurs when the plasmonic structure is complementary to an annulus of constant, isotropic medium.
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 particula
r, 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.
This is a survey of approximate cloaking using transformation optics for acoustic and electromagnetic waves.
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 ex
plicit. 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.
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.
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 $epsilon$ in three dimensions and $|lnepsilon|^{-1}$ in two dimensions, where $epsilon$ is the regularization parameter.
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 pre
cisely, 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.
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.
We establish improv
