We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation (−∆) s u + q(x, u) = 0 with s ∈ (0, 1). We show that an unknown function q(x, u) can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to 2. Moreover, we demonstrate the comparison principle and provide a L ∞ estimate for this nonlocal equation under appropriate regularity assumptions.

Abstract. In this work we consider stability of recovery of the conductivity and attenuation coefficients of the stationary Maxwell and Schrödinger equations from a complete set of (Cauchy) boundary data. By using complex geometrical optics solutions we derive some bounds which can be viewed as an evidence of increasing stability in these inverse problems when frequency is growing.

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Abstract. We consider the problem of reconstructing the features of a weak anisotropic background potential by the trajectories of vortex dipoles in a nonlinear Gross-Pitaevskii equation. At leading order, the dynamics of vortex dipoles are given by a Hamiltonian system. If the background potential is sufficiently smooth and flat, the background can be reconstructed using ideas from the boundary and the lens rigidity problems. We prove that reconstructions are unique, derive an approximate reconstruction formula, and present numerical examples.

is paper concerns an hybrid inverse problem involving elastic measurements called Transient Elastography (TE) which enables detection and characterization of tissue abnormalities. In this paper we assume that the displacements are modeled by linear isotropic elasticity system and the tissue displacement has been obtained by the rst step in hybrid methods. We reconstruct Lamé parameters of this system from knowledge of the tissue displacement. More precisely, we show that for a su ciently large number of solutions of the elasticity system and for an open set of the well-chosen boundary conditions, Lamé parameters can be uniquely and stably reconstructed. e set of well-chosen boundary conditions is characterized in terms of appropriate complex geometrical optics solutions.