Eric Bonnetier

The Neumann-Poincar\'e (NP) operator naturally appears in the context of metamaterials as it may be used to represent the solutions of elliptic transmission problems via potentiel theory. In particular, its spectral properties are closely related to the well-posedness of these PDE's, in the typical case where one considers a bounded inclusion of homogeneous plasmonic metamaterial embedded in a homogeneous background dielectric medium. In a recent work, M. Perfekt and M. Putinar have shown that the NP operator of a 2D curvilinear polygon has an essential spectrum, which depends only on the angles of the corners. Their proof is based on quasi-conformal mappings and techniques from complex-analysis. In this work, we characterize the spectrum of the NP operator for a 2D domain with corners in terms of elliptic corner singularity functions, which gives insight on the behavior of generalized eigenmodes.

This paper is devoted to superlensing using hyperbolic metamaterials: the possibility to image an arbitrary object using hyperbolic metamaterials without imposing any conditions on size of the object and the wave length. To this end, two types of schemes are suggested and their analysis are given. The superlensing devices proposed are independent of the object. It is worth noting that the study of hyperbolic metamaterials is challenging due to the change type of modelling equations, elliptic in some regions, hyperbolic in some others.

This paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography. We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.

In this paper we derive the asymptotic expansion of the velocity field of a small deformable droplet immersed in an incompressible Newtonian fluid. Using an appropriate physical scaling of the surface tension with respect to the droplet volume, we show that the first order of the asymptotic can be expressed in terms of the velocity field in the absence of the droplet and a new kind of moment tensor, called the curvature moment tensor. Our asymptotic formula extends those already derived for rigid droplets and aimed to obtain simplified macroscale properties of a dilute suspension composed of identical droplets dispersed in an incompressible Newtonian fluid from knowledge of its microscopic properties. We finally determine explicitly the curvature moment tensor for ellipses and ellipsoids.

This paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography. We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.