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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.

We discuss the stability issue for Calderón's inverse conductivity problem, also known as Electrical Impedance Tomography. It is well known that this problem is severely ill-posed. In this paper we prove that if it is a-priori known that the conductivity is piecewise constant with a bounded number of unknown values, then a Lipschitz stability estimate holds.

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We study mappings from R 2 into R 2 whose components are weak solutions to the elliptic equation in divergence form, Div(σ ∇u) = 0, which we call σ -harmonic mappings. We prove sufficient conditions for the univalence, i.e., injectivity, of such mappings. Moreover we prove local bounds in BMO on the logarithm of the Jacobian determinant of such univalent mappings, thus obtaining the a.e. nonvanishing of their Jacobian. In particular, our results apply to σ -harmonic mapping associated with any periodic structure and therefore they play an important role in homogenization.