We derive different classes of generalized impedance boundary conditions for the scattering problem from strongly absorbing obstacles. Compared to existing works, our construction is based on an asymptotic development of the solution with respect to the medium absorption. Error estimates are derived to validate the accuracy of each condition.

Using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time-varying feedback laws. To this end, on the one hand, we provide a new proof of the wellposedness and the "optimal" bound with respect to damping constants for the solutions of the kernel equations; this allows one to deal with variable coefficients, even with a weak regularity of these coefficients. On the other hand, we establish the well-posedness and estimates for the heat equations with a nonlocal boundary condition at one side. Mathematics Subject Classification

No abstract

Abstract. This paper is devoted to the study of the behavior of the unique solutionwhere Ω is a smooth connected bounded open subset of, k is a non-negative constant, A is a uniformly elliptic matrixvalued function, Σ is a real function bounded above and below by positive constants, and s δ is a complex function whose real part takes the values 1 and −1 and whose imaginary part is positive and converges to 0 as δ goes to 0. This is motivated from a result of Nicorovici, McPhedran, and Milton; another motivation is the concept of complementary media. 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.

We present new results concerning the approximation of the total variation, |∇u|, of a function u by non-local, non-convex functionals of the formwhere is a domain in R d and ϕ : [0, +∞) → [0, +∞) is a nondecreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi's concept ofconvergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing.

In this paper, we present some new characterizations of Sobolev spaces. Here is a typical result. Let

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 as the loss goes to 0, 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.

Cloaking a source via anomalous localized resonance (ALR) was discovered by Milton and Nicorovici in [14]. 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 [19] 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 [19]. 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 [19] 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 arbitrary source outside the core-shell structure in the doubly complementary media setting.