Section: New Results

Inverse problems for heterogeneous systems

In [7] , David Dos Santos Ferreira et al. obtain global stability estimates for a potential in a Schrödinger equation on an open bounded set in dimension $n=3$ from the Dirichlet-to-Neumann map with partial data. This improves previous results where local stability was proved : the region under control was the penumbra delimited by a source of light outside of the convex hull of the open set. These local estimates provided stability of log-log type corresponding to the uniqueness results in Calderón's inverse problem with partial data proved by Kenig, Sjöstrand and Uhlmann. The corresponding global estimates are proved in all dimensions higher than three. The estimates are based on the construction of solutions of the Schrödinger equation by complex geometrical optics developed in the anisotropic setting by Dos Santos Ferreira, Kenig, Salo and Uhlmann to solve the Calderón problem in certain admissible geometries.

In [20] , David Dos Santos Ferreira et al. proved uniform $L^p$ resolvent estimates for the stationary damped wave operator. Uniform $L^p$ resolvent estimates for the Laplace operator on a compact smooth Riemannian manifold without boundary were first established by Shen on the Torus, then by Dos Santos Ferreira-Kenig-Salo for general compact manifolds and advanced further by Bourgain-Shao-Sogge-Yao. An alternative proof relying on the techniques of semiclassical Strichartz estimates allows to handle non-self-adjoint perturbations of the Laplacian and embeds very naturally in the semiclassical spectral analysis framework, and applies in the damped wave context.

In [10] , Munnier and Ramdani considered the 2D inverse problem of recovering the positions and the velocities of slowly moving small rigid disks in a bounded cavity filled with a perfect fluid. Using an integral formulation, they first derive an asymptotic expansion of the DtN map of the problem as the diameters of the disks tend to zero. Then, combining a suitable choice of exponential type data and the DORT method (french acronym for Diagonalization of the Time Reversal Operator), a reconstruction method for the unknown positions and velocities is proposed. Let us emphasize here that this reconstruction method uses in the context of fluid-structure interaction problems a method which is usually used for waves inverse scattering (the DORT method).

In [24] , Munnier and Ramdani proposed a new method to tackle a geometric inverse problem related to Calderón's inverse problem. More precisely, they proposed an explicit reconstruction formula for the cavity inverse problem using conformal mapping. This formula is derived by combining two ingredients: a new factorization result of the DtN map and the so-called generalized Polia-Szegö tensors of the cavity.

In [11] , Ramdani, Tucsnak and Valein tackled a state estimation problem for an infinite dimensional system arising in population dynamics (a linear model for age-structured populations with spatial diffusion). Assume the initial state to be unknown, the considered inverse problem is to estimate asymptotically on time the state of the system from a locally distributed observation in both age and space. This is done by designing a Luenberger observer for the system, taking advantage of the particular spectral structure of the problem (the system has a finite number of unstable eigenvalues).

In [12] , San Martin, Schwindt and Takahashi consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. They deal with the case where the fluid equations are the non stationary Stokes system and using the enclosure method, they can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, they can obtain the same result in the case of a linear fluid–structure system composed by a rigid body and a viscous incompressible fluid. They also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid–structure system with free boundary. Using complex spherical waves, they obtain some partial information on the distance from a point to the obstacle.