Cloaking via anomalous localized resonance for doubly complementary media in the finite frequency regime

Cloaking a source via anomalous localized resonance (ALR) was discovered by Milton and Nicorovici in [15]. A general setting in which cloaking a source via ALR takes place is the setting of doubly complementary media. This was introduced and studied in [20] for the quasistatic regime. In this paper, we study cloaking a source via ALR for doubly complementary media in the finite frequency regime. To this end, 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, on one hand we extend ideas from [20] and on the other hand we add new insights into the problem. This allows us not only to overcome difficulties related to the finite frequency regime but also 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

Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients

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 [26], 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. Three approaches are proposed to obtain the a priori estimates. The first one follows from a priori estimates of elliptic systems equipped with complementing boundary conditions due to Agmon, Douglis, and Nirenberg in their classic work [1]. The second approach, which complements the first one, is variational and based on the Dirichlet principle. The last approach, which complements the second one, is also variational and uses the multiplier technique. Using these approaches, we are able to obtain new results on the well-posedness of these equations for which the conditions on the coefficients are imposed "partially" or "not strictly" on the interfaces of sign changing coefficients. This allows us to rediscover and extend 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 not in H-loc(1)(R-d) as usual and possibly not even in L-loc(2)(R-d). The optimality of our results is also discussed. (C) 2016 Elsevier Masson SAS. All rights reserved

Estimates for the topological degree and related topics

This is a survey paper on estimates for the topological degree and related topics which range from the characterizations of Sobolev spaces and BV functions to the Jacobian determinant and nonlocal filter problems in Image Processing. These results are obtained jointly with Bourgain and Brezis. Several open questions are mentioned

Cloaking using complementary media in the quasistatic regime

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. (C) 2015 Elsevier Masson SAS. All rights reserved