Strong unique continuation for general elliptic equations in 2D

We prove that solutions to elliptic equations in two variables in divergence form, possibly non-selfadjoint and with lower order terms, satisfy the strong unique continuation property.Comment: 10 page

Corrigendum to ``Determining a sound-soft polyhedral scatterer by a single far-field measurement''

In the paper, G. Alessandrini and L. Rondi, ``Determining a sound-soft polyhedral scatterer by a single far-field measurement'', Proc. Amer. Math. Soc. 133 (2005), pp. 1685-1691, on the determination of a sound-soft polyhedral scatterer by a single far-field measurement, the proof of Proposition 3.2 is incomplete. In this corrigendum we provide a new proof of the same proposition which fills the previous gap.Comment: 3 page

Cracks with impedance, stable determination from boundary data

We discuss the inverse problem of determining the possible presence of an (n-1)-dimensional crack \Sigma in an n-dimensional body \Omega with n > 2 when the so-called Dirichlet-to-Neumann map is given on the boundary of \Omega. In combination with quantitative unique continuation techniques, an optimal single-logarithm stability estimate is proven by using the singular solutions method. Our arguments also apply when the Neumann-to-Dirichlet map or the local versions of the D-N and the N-D map are available.Comment: 40 pages, submitte

Single-logarithmic stability for the Calder\'on problem with local data

We prove an optimal stability estimate for Electrical Impedance Tomography with local data, in the case when the conductivity is precisely known on a neighborhood of the boundary. The main novelty here is that we provide a rather general method which enables to obtain the H\"older dependence of a global Dirichlet to Neumann map from a local one on a larger domain when, in the layer between the two boundaries, the coefficient is known.Comment: 12 page

Quantitative estimates on Jacobians for hybrid inverse problems

We consider $\sigma$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u_i)=0$, for $i=1,\ldots,n$. We investigate whether, with suitably prescribed Dirichlet data, the Jacobian determinant can be bounded away from zero. Results of this sort are required in the treatment of the so-called hybrid inverse problems, and also in the field of homogenization studying bounds for the effective properties of composite materials.Comment: 15 pages, submitte

Determining the anisotropic traction state in a membrane by boundary measurements

We prove uniqueness and stability for an inverse boundary problem associated to an anisotropic elliptic equation arising in the modeling of prestressed elastic membranes.Comment: 6 page

Invertible harmonic mappings, beyond Kneser

We prove necessary and sufficient criteria of invertibility for planar harmonic mappings which generalize a classical result of H. Kneser, also known as the Rad\'{o}-Kneser-Choquet theorem.Comment: One section added. 15 page

Estimates for the dilatation of σ\sigma-harmonic mappings

We consider planar $\sigma$-harmonic mappings, that is mappings $U$ whose components $u^1$ and $u^2$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u^i)=0$, for $i=1,2$. We investigate whether a locally invertible $\sigma$-harmonic mapping $U$ is also quasiconformal. Under mild regularity assumptions, only involving $\det \sigma$ and the antisymmetric part of $\sigma$, we prove quantitative bounds which imply quasiconformality.Comment: 8 pages, to appear on Rendiconti di Matematica e delle sue applicazion