Giovanni Alessandrini

Stable determination of conductivity by boundary measurements

We consider the problem of determining the scalar coefficient γ in the elliptic equation div(γ grad u) = 0 in ω when, for every Dirichlet datum u = ∅ on ∂ω , the Neumann datum γ(∂/ ∂ n)u = ∧.γ∅ is known. We prove a continuous dependence result

The stability for the Cauchy problem for elliptic equations

Methods for forming thin layer barrier layer films for use in enzyme containing laminated membranes and membranes formed thereby are disclosed. The barrier layers exhibit improved acetaminophen rejection and comprise a cellulose acetate/cellulose acetate butyrate blend. The thin layer barrier membranes are formed from a plural solvent containing solution and are cured at a critical temperature of about 102 DEG -114 DEG F., most preferably at about 106 DEG F.-114 DEG F. while traveling through a circulating hot air oven. Alternatively, the membranes can be cured at room temperature or in a stagnant oven at temperatures of from room temperature to about 175 DEG C. (350 DEG F.) for a period of from about 10 minutes to 1 hour.

Singular solutions of elliptic equations and the determination of conductivity by boundary measurements

Abstract We improve some results on uniqueness and stability in the determination of the coefficient a in the equation (i) div( a grad u ) = 0 in Ω, when all possible pairs of Dirichlet and Neumann data on ∂Ω are known. We also treat cases of anisotropic equations. Our method relies on the construction of solutions to (i) having an isolated singularity with prescribed asymptotic behaviour.

Stable determination of a crack from boundary measurements

We treat the problem of determining a crack inside a conductor when two pairs of current and voltage boundary measurements are given. We prove a theorem of continuous dependence from the data.

Unique Determination of Multiple Cracks by Two Measurements

We study the inverse problem of determining multiple cracks in a planar conductor by electrostatic measurements at the boundary. We prove that two measurements at the boundary suffice to identify multiple cracks with any number of components. We treat the problem under no regularity assumptions on the cracks and on the background conductivity.

Univalent σ-harmonic mappings

Abstract: We study mappings from ℝ2 into ℝ2 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.

Stable determination of corrosion by a single electrostatic boundary measurement

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

Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions

The Stekloff eigenvalue problem (1.1) has a countable number of eigenvalues

EXAMPLES OF INSTABILITY IN INVERSE BOUNDARY-VALUE PROBLEMS

\{ p_n \} n = 1,2, \ldots

Optimal size estimates for the inverse conductivity problem with one measurement

, each of finite multiplicity. In this paper the authors give an upper estimate, in terms of the integer n, of the multiplicity of