Elliptic and Parabolic Equations - Abstract
Bonnetier, Eric
This work connects to a series of experiments that showed super-resolution in time-reversal experiments in structured media. We consider a set of small scattering inclusions distributed in a homogenous dielectric medium in the plane. The scatterers are contained in a bounded region $Omega$, in which they are periodically distributed, with a period $varepsilon$. We assume that $Omega$ also contains a defect, in the form of a subset $omega_d$, of size $O(varepsilon)$, filled with another dielectric material. We asymptotically compare the solutions $u_varepsilon$ and $u_varepsilon ,d$ to the Helmholtz equation, respectively in the absence or in the presence of the defect. We show that as $varepsilon rightarrow 0$, the difference $u_varepsilon_d - u_varepsilon$ far from the defect is proportional to the gradient of the Green's function of the em homogenized medium, obtained as the period of the scatterers tends to 0. In other words, the defect can be detected with a resolution corresponding to a medium, which depends on the material parameters of the scatterers: Their choice may improve the resolution from that of the surrounding homogeneous medium. The main ingredients in this analysis, are uniform $W^1,infty$ estimates for the solutions of elliptic PDE's in periodic media.
Common work with Habib Ammari (CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France. ammari@cmapx.polytechnique.fr) and Yves Capdeboscq (Mathematical Institute, University of Oxford, Oxford OX1 3LB, U.K. yves.capdeboscq@maths.ox.ac.uk)