arXiv subjects
Publications and source records attributed to Elisa Francini.
In this paper we consider the problem of determining an unknown pair $λ$, $μ$ of piecewise constant Lamé parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $λ$ and $μ$ from the Dirichlet to Neumann map.
In this paper we estimate the size of a measurable inclusion in terms of power measurements for a single applied boundary current. This problem arises in medical imaging for the screening of organs. For this kind of problem one has to deal mathematically with the complex conductivity (admittivity) equation. In this case we are able to establish, for certain classes of admittivities, lower and upper bounds of the measure of the inclusion in terms of the power measurements. A novelty of our result is that we are able to estimate also the volume of inclusions having part of its boundary in common with the reference body. Our analysis is based on the derivation of energy bounds and of fine quantitative estimates of unique continuation for solutions to elliptic equations.
We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in the presence of small inhomogeneities. Both the body and the inclusions are allowed to be anisotropic. This work extends prior work of CapdeBoscq and Vogelius ({\em Math. Modelling Num. Anal.} 37, 2003) for the conductivity case. In particular, we obtain an asymptotic expansion of the difference between the displacements at the boundary with and without inclusions, under Neumann boundary conditions, to first order in the measure of the inclusions. We impose no geometric conditions on the inclusions, which need only be measurable sets. The first-order correction contains an elastic moment tensor $\MM$ that encodes the effect of the inclusions. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for $\MM$ only in terms of the elasticity tensors, which we assume strongly convex, their inverses, and a frame on the curve that supports the inclusion. We prove uniqueness of $\MM$ in this setting and recover the formula previously obtained by Beretta and Francini ({\em SIAM J. Math. Anal.}, 38, 2006).
In this paper we investigate the boundary value problem ${div(γ\nabla u)=0 in Ω, u=f on \partialΩ$ where $γ$ is a complex valued $L^\infty$ coefficient, satisfying a strong ellipticity condition. In Electrical Impedance Tomography, $γ$ represents the admittance of a conducting body. An interesting issue is the one of determining $γ$ uniquely and in a stable way from the knowledge of the Dirichlet-to-Neumann map $Λ_γ$. Under the above general assumptions this problem is an open issue. In this paper we prove that, if we assume a priori that $γ$ is piecewise constant with a bounded known number of unknown values, then Lipschitz continuity of $γ$ from $Λ_γ$ holds.