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.

Abstract. We obtain quantitative estimates of unique continuation for solutions to parabolic equations: doubling properties and two-sphere one-cylinder inequalities.

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In this paper we will review the main results concerning the issue of stability for the determination unknown boundary portion of a thermic conducting body from Cauchy data for parabolic equations. We give detailed and selfcontained proofs. We prove that such problems are severely ill-posed in the sense that under a priori regularity assumptions on the unknown boundaries, up to any finite order of differentiability, the continuous dependence of unknown boundary from the measured data is, at best, of logarithmic type.We review the main results concerning quantitative estimates of unique continuation for solutions to second order parabolic equations. We give a detailed proof of a Carleman estimate crucial for the derivation of the stability estimates.