This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance...

Les données, i.e. l’ouvert $\Omega$ et la force appliquée $f$, sont supposées de classe $𝒞$. Il est montré que toute solution des équations de Navier-Stokes dans l’ouvert $\Omega$, bornée dans $H^1(\Omega {)}^N$ ($N=2$ ou $3$) sur un intervalle de temps semi-infini $(t_0+\infty )$, est aussi bornée, pour $t\to +\infty$, dans tous les espaces $H^m(\Omega {)}^N$. Il en résulte que tout ensemble fonctionnel invariant ou attracteur borné dans $H^1\left(\Omega \right)(N$ (ou même $H^{1/2+ϵ}(\Omega {)}^N$, $ϵ\>0$) est porté par ${𝒞}^{\infty }(\bar{\Omega })$. Le cas où les forces appliquées dérivent d’un potentiel (i.e. $f=0$) est abordé : il est montré que toute solution...

In the paper definitions of various kinds of stability and boundedness of solutions of linear controllable systems of partial differential equations are introduced and their interconnections are derived. By means of Ljapunov's functions theorems are proved which give necessary and sufficient conditions for particular kinds of stability and boundedness of the solutions.

We consider the Cauchy problem in an unbounded region for equations of the type either $D_{t}z(t,x)=f(t,x,z(t,x),z_{(t,x)},D_{x}z(t,x))$ or $D_{t}z(t,x)=f(t,x,z(t,x),z,D_{x}z(t,x))$. We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.