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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.

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.

The Stekloff eigenvalue problem (1.1) has a' countable number of eigenvalues (Pn }n= 1,2. .. .. each of finite multiplicity. In this paper the authors give an upper estimate, in terms of the integer n, of the multiplicity of Pn, and the number of critical points and of nodal domains of the eigenfunctions corresponding to Pn. In view of a possible application to inverse conductivity problems, the result for the general case of elliptic equations with discontinuous coefficients in divergence form is proven by replacing the classical concept of critical point with the more suitable notion of geometric critical point.

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