Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities

We consider the stability issue of the inverse conductivity problem for a conformal class of anisotropic conductivities in terms of the local Dirichlet\u2013 Neumann map. We extend here the stability result obtained by Alessandrini and Vessella (Alessandrini G and Vessella S 2005 Lipschitz stability for the inverse conductivity problem Adv. Appl. Math. 35 207\u2013241), where the authors considered the piecewise constant isotropic case

A transmission problem on a polygonal partition: regularity and shape differentiability

We consider a transmission problem on a polygonal partition for the two-dimensional conductivity equation. For suitable classes of partitions we establish the exact behaviour of the gradient of solutions in a neighbourhood of the vertexes of the partition. This allows to prove shape differentiability of solutions and to establish an explicit formula for the shape derivative

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

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

Instability in the Gel'fand inverse problem at high energies

We give an instability estimate for the Gel'fand inverse boundary value problem at high energies. Our instability estimate shows an optimality of several important preceeding stability results on inverse problems of such a type

A Lipschitz stable reconstruction formula for the inverse problem for the wave equation

We consider the problem to reconstruct a wave speed c ∈ C∞(M) in a domain M ⊂ R n from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map Λ. We introduce a reconstruction formula for c that is based on the Boundary Control method and incorporates features also from the complex geometric optics solutions approach. Moreover, we show that the reconstruction formula is locally Lipschitz stable for a low frequency component of c −2 under the assumption that the Riemannian manifold (M, c−2dx2 ) has a strictly convex function with no critical points. That is, we show that for all bounded C 2 neighborhoods U of c, there is a C 1 neighborhood V of c and constants C, R > 0 such that |F ec −2 − c −2 � (ξ)| ≤ Ce2R|ξ| Λe − Λ ∗ , ξ ∈ R n , for all ec ∈ U ∩ V , where Λ is the Dirichlet-to-Neumann map corre- e sponding to the wave speed ec and k·k∗ is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps