We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations.We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions.As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.

Abstract. We prove that a sound-soft polyhedral scatterer is uniquely determined by the far-field pattern corresponding to an incident plane wave at one given wavenumber and one given incident direction.Lo duca e io per quel cammino ascoso intrammo a ritornar nel chiaro mondo; e sanza cura aver d'alcun riposo, salimmo su, el primo e io secondo, tanto ch'i' vidi de le cose belle che porta'l ciel, per un pertugio tondo; e quindi uscimmo a riveder le stelle.Dante, Inferno, C.XXXIV, 133-139.

We prove an optimal stability estimate for an inverse Robin boundary value problem arising in corrosion detection by electrostatic boundary measurements.

Abstract.We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is well-known that this problem is highly illposed. In this work, we propose the use of the Mumford-Shah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several numerical examples. Our results indicate that this is an effective approach for overcoming the illposedness. Moreover, it has the capability of enhancing the reconstruction while at the same time segmenting the conductivity image.

Following a recent paper by N. Mandache (Inverse Problems 17 (2001), pp. 1435-1444), we establish a general procedure for determining the instability character of inverse problems. We apply this procedure to many elliptic inverse problems concerning the determination of defects of various types by different kinds of boundary measurements and we show that these problems are exponentially ill-posed.

We study the stability for the direct acoustic scattering problem with sound-hard scatterers with minimal regularity assumptions on the scatterers. The main tool we use for this purpose is the convergence in the sense of Mosco.\ud \ud We obtain uniform decay estimates for scattered fields and we investigate how a sound-hard screen may be approximated by thin sound-hard obstacles

In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter α ∈ R. The case α = 0 corresponds to purely logarithmic interactions, minimised by the celebrated circle law for a quadratic confinement; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for α ∈ (0, 1) the minimiser can be computed explicitly and is the normalised characteristic function of the domain enclosed by an ellipse. To prove our result we borrow techniques from fluid dynamics, in particular those related to Kirchhoff's celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses. Therefore we show a surprising connection between vortices and dislocations.

This paper is concerned with the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium.We prove a sharp stability result for the solutions to the direct electromagnetic scattering problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. The stability result leads to bounds on solutions to the scattering problems which are uniform for an extremely general class of admissible scatterers and inhomogeneities.These uniform bounds are a key step to tackle the challenging stability issue for the corresponding inverse electromagnetic scattering problem. In this paper we establish two optimal stability results of logarithmic type for the determination of polyhedral scatterers by a minimal number of electromagnetic scattering measurements.In order to prove the stability result for the direct electromagnetic scattering problem, we study two fundamental issues in the theory of Maxwell equations: Mosco convergence for H(curl) spaces and higher integrability properties of solutions to Maxwell equations in nonsmooth domains.