Luisa Faella

In [2], devoted to solving the inverse obstacle problem for non- linear materials via the Monotonicity Principle, two issues were found in the for- mulation of some hypotheses.

The Monotonicity Principle (MP), stating a monotonic relationship between a material property and a proper corresponding boundary operator, is attracting great interest in the field of inverse problems, because of its fundamental role in developing real time imaging methods. Moreover, under quite general assumptions, a MP for elliptic PDEs with nonlinear coefficients has been established. This MP provided the basis for introducing a new imaging method to deal with the inverse obstacle problem, in the presence of nonlinear anomalies. This constitutes a relevant novelty because there is a general lack of quantitative and physic based imaging method, when nonlinearities are present. The introduction of a MP based imaging method poses a set of fundamental questions regarding the performance of the method in the presence of noise. The main contribution of this work is focused on theoretical aspects and consists in proving that (i) the imaging method is stable and robust with respect to the noise, (ii) the reconstruction approaches monotonically to a well-defined limit, as the noise level approaches to zero, and that (iii) the limit contains the unknown set and is contained in the outer boundary of the unknown set. Results (i) and (ii) come directly from the MP, while result (iii) requires to prove the so-called Converse of the MP, a theoretical result of fundamental relevance to evaluate the ideal (noise-free) performances of the imaging method. The results are provided in a quite general setting for Calderón problem, and proved for three wide classes where the nonlinearity of the anomaly can be either bounded from infinity and zero, or bounded from zero only, or bounded by infinity only. These classes of constitutive relationships cover the wide majority of cases encountered in applications.

This paper deals with the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponent. The results obtained are relevant because they enable the use of a fast imaging method based on MP, applied to a wide class of problems with two or more materials, at least one of which is nonlinear. The treatment is very general and makes it possible to model a wide range of practical configurations such as superconducting (SC), perfect electrical conducting (PEC) or perfect electrical insulating (PEI) materials. A key role is played by the average Dirichlet-to-Neumann operator, introduced in Corbo Esposito et al (2021 Inverse Problems 37 045012), where the MP for a single type of nonlinearity was treated. Realistic numerical examples confirm the theoretical findings.

This paper presents a method of transforming from the three-dimensional piezoelastic problem for a composite material to the one-dimensional problem for a piezoelastic beam. This is done using the asymptotic homogenization technique based on the separation of fast and slow variables in the solution. A special feature of the problem is the presence of two small parameters, one of which characterizes the microstructure of the composite material, and the other defines the cross-sectional size. Homogenized relations describing the piezoelastic beam and fast correctors were obtained. Their joint use makes it possible to correctly describe the total stress-strain state of the original three-dimensional body. The proposed method is suitable for solving the three-dimensional problem of deformation of an extended body with an arbitrary periodic structure as well as for solving new problems (e.g., the torsion problem) that have no analogues in the theory of piezoelastic plates.

In this paper, starting from a non-convex and nonlocal $3D-$variational model for the electric polarization in a ferroelectric material, and using an asymptotic process based on dimensional reduction, we analyze junction phenomena for two vertical joined ferroelectric thin cilynders with differents small thicknesses. We obtain three variational models for thin wire joined to a thin film, depending on the limit of the ratio between these two thicknesses.