Small volume asymptotics for a defect in a periodic medium and enhancement of the resolution

Eric Bonnetier (1)

1 EDP - Equations aux Dérivées Partielles

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. [e-mail]) and Yves Capdeboscq (Mathematical Institute, University of Oxford, Oxford OX1 3LB, U.K. [e-mail])

Domaines: Equations aux dérivées partielles [math.AP]

Eric Bonnetier. Small volume asymptotics for a defect in a periodic medium and enhancement of the resolution. EPE 09 - International Conference on Elliptic and Parabolic Equations, Nov 2009, Berlin, Germany. ⟨hal-00750517⟩