arXiv subjects

Edi Rosset

Publications and source records attributed to Edi Rosset.

At least 19 recordsLinked to original sources

arXiv2026

Identification of an inclusion from local Cauchy data for time-harmonic elastic waves

We consider the inverse problem of determining an inclusion contained in an elastic body undergoing time-harmonic oscillations at fixed frequency by local Cauchy data. Both the body and the inclusion are made by different homogeneous linearly elastic isotropic materials, with piecewise constant mass densities. Under mild a priori regularity assumptions on the unknown inclusion and with no spectral hypothesis on the frequency, we establish a logarithmic-type stability estimate by local Cauchy data. We also provide a stability result in terms of the so local Dirichlet to Neumann map for sufficiently small frequency.

math.AP↗

arXiv2025

Lipschitz Stability for Polyhedral Elastic Inclusions from Partial Data

The paper deals with the inverse problem of determining a polyhedral inclusion compactly contained in an elastic body from boundary measurements of traction and displacement taken on an open portion of the boundary. Both the inclusion and the body are made of homogeneous isotropic material. Under suitable assumptions on the geometry of the unknown inclusion, we prove a constructive Lipschitz stability estimate from the local Dirichlet-to-Neumann map.

math.AP↗

arXiv2022

Strong unique continuation and global regularity estimates for nanoplates

In this paper we analyze some properties of a sixth order elliptic operator arising in the framework of the strain gradient linear elasticity theory for nanoplates in flexural deformation. We first rigorously deduce the weak formulation of the underlying Neumann problem as well as its well posedness. Under some suitable smoothness assumptions on the coefficients and on the geometry we derive interior and boundary regularity estimates for the solution of the Neumann problem. Finally, for the case of isotropic materials, we obtain new Strong Unique Continuation results in the interior, in the form of doubling inequality and three spheres inequality, by a Carlemann estimates approach.

math.AP↗

arXiv2022

Size estimates for nanoplates

We consider the problem of determining, within an elastic isotropic nanoplate in bending, the possible presence of an inclusion made of different elastic material. Under suitable a priori assumptions on the unknown inclusion, we provide quantitative upper and lower estimates for the area of the unknown defect in terms of the works exerted by the boundary data when the inclusion is present or absent.

math.AP↗

arXiv2021

Doubling inequality at the boundary for the Kirchhoff-Love plate's equation with supported conditions

In this article we derive a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate's equation satisfying supported boundary conditions. To this end, we combine the use of a suitable conformal mapping which flattens the boundary and a reflection argument which guarantees the needed regularity of the extended solution. We finally apply inequalities of Carleman type in order to derive the result. The latter implies Strong Unique Continuation Property at the boundary (SUCPB).

math.AP↗

arXiv2020

Estimating the area of extreme inclusions in Reissner-Mindlin plates

We derive upper and lower estimates of the area of unknown defects in the form of either cavities or rigid inclusions in Mindlin-Reissner elastic plates in terms of the difference $\delta W$ of the works exerted by boundary loads on the defected and on the reference plate. It turns out that the upper estimates depend linearly on $\delta W$, whereas the lower ones depend quadratically on $\delta W$. These results continue a line of research concerning size estimates of extreme inclusions in electric conductors, elastic bodies and plates.

math.AP↗

arXiv2020

Optimal identification of cavities in the Generalized Plane Stress problem in linear elasticity

For the Generalized Plane Stress (GPS) problem in linear elasticity, we obtain an optimal stability estimate of logarithmic type for the inverse problem of determining smooth cavities inside a thin isotropic cylinder {}from a single boundary measurement of traction and displacement. The result is obtained by reformulating the GPS problem as a Kirchhoff-Love plate-like problem in terms of the Airy's function, and by using the strong unique continuation at the boundary for a Kirchhoff-Love plate operator under homogeneous Dirichlet conditions, which has been recently obtained in \cite{l:arv}.

math.AP↗

arXiv2019

Doubling Inequality at the Boundary for the Kirchhoff -- Love Plate's Equation with Dirichlet Conditions

The main result of this paper is a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate's equation satisfying homogeneous Dirichlet conditions. This result, like the three sphere inequality with optimal exponent at the boundary proved in Alessandrini, Rosset, Vessella, Arch. Ration. Mech. Anal. (2019), implies the Strong Unique Continuation Property at the Boundary (SUCPB). Our approach is based on a suitable Carleman estimate, and involves an ad hoc reflection of the solution. We also give a simple application of our main result, by weakening the standard hypotheses ensuring uniqueness for the Cauchy problem for the plate equation.

math.AP↗

arXiv2018

Optimal stability in the identification of a rigid inclusion in an isotropic Kirchhoff-Love plate

In this paper we consider the inverse problem of determining a rigid inclusion inside a thin plate by applying a couple field at the boundary and by measuring the induced transversal displacement and its normal derivative at the boundary of the plate. The plate is made by non-homogeneous, linearly elastic and isotropic material. Under suitable a priori regularity assumptions on the boundary of the inclusion, we prove a constructive stability estimate of log type. Key mathematical tool is a recently proved optimal three spheres inequality at the boundary for solutions to the Kirchhoff-Love plate's equation.

math.AP↗

arXiv2018

Optimal three spheres inequality at the boundary for the Kirchhoff-Love plate's equation with Dirichlet conditions

We prove a three sphere inequality with optimal exponent at the boundary for solutions to the Kirchhoff-Love plate's equation satisfying homogeneous Dirichlet conditions. This result implies the Strong Unique Continuation Property at the Boundary (SUCPB). Our approach is based on the method of Carleman estimates, and involves the construction of an ad hoc conformal mapping preserving the structure of the operator and the employment of a suitable reflection of the solution with respect to the flatten boundary which ensures the needed regularity of the extended solution. To the authors' knowledge, this is the first (nontrivial) SUCPB result for fourth-order equations with bi-Laplacian principal part.

math.AP↗

arXiv2017

Size estimates for fat inclusions in an isotropic Reissner-Mindlin plate

In this paper we consider the inverse problem of determining, within an elastic isotropic thick plate modelled by the Reissner-Mindlin theory, the possible presence of an inclusion made of a different elastic material. Under some a priori assumptions on the inclusion, we deduce constructive upper and lower estimates of the area of the inclusion in terms of a scalar quantity related to the work developed in deforming the plate by applying simultaneously a couple field and a transverse force field at the boundary of the plate. The approach allows to consider plates with boundary of Lipschitz class.

math.AP↗

arXiv2017

On an elastic model arising from volcanology: an analysis of the direct and inverse problem

In this paper we investigate a mathematical model arising from volcanology describing surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. The modeling assumptions translate mathematically into a Neumann boundary value problem for the classical Lam\'e system in a half-space with an embedded pressurized cavity. We establish well-posedness of the problem in suitable weighted Sobolev spaces and analyse the inverse problem of determining the pressurized cavity from partial measurements of the displacement field proving uniqueness and stability estimates.

math.AP↗

arXiv2016

A generalized Korn inequality and strong unique continuation for the Reissner-Mindlin plate system

We prove constructive estimates for elastic plates modelled by the Reissner-Mindlin theory and made by general anisotropic material. Namely, we obtain a generalized Korn inequality which allows to derive quantitative stability and global H^2 regularity for the Neumann problem. Moreover, in case of isotropic material, we derive an interior three spheres inequality with optimal exponent from which the strong unique continuation property follows.

math.AP↗

arXiv2016

Stable determination of an inclusion in an inhomogeneous elastic body by boundary measurements

In this paper we consider the stability issue for the inverse problem of determining an unknown inclusion contained in an elastic body by all the pairs of measurements of displacement and traction taken at the boundary of the body. Both the body and the inclusion are made by inhomogeneous linearly elastic isotropic material. Under mild a priori assumptions about the smoothness of the inclusion and the regularity of the coefficients, we show that the logarithmic stability estimate proved in \cite{ADiCMR14} in the case of piecewise constant coefficients continues to hold in the inhomogeneous case. We introduce new arguments which allow to simplify some technical aspects of the proof given in \cite{ADiCMR14}.

math.AP↗

arXiv2015

Global stability for an inverse problem in soil-structure interaction

We consider the inverse problem of determining the Winkler subgrade reaction coefficient of a slab foundation modelled as a thin elastic plate clamped at the boundary. The plate is loaded by a concentrated force and its transversal deflection is measured at the interior points. We prove a global H\"{o}lder stability estimate under (mild) regularity assumptions on the unknown coefficient.

math.AP↗

arXiv2014

Lipschitz continuous dependence of piecewise constant Lam\'e coefficients from boundary data: the case of non flat interfaces

We consider the inverse problem of determining the Lam\'e moduli for a piecewise constant elasticity tensor ${\mathbb C}= \sum_{j} {\mathbb C}_j \chi_{D_j}$, where $\{D_j\}$ is a known finite partition of the body $\Omega$, from the Dirichlet-to-Neumann map. We prove that Lipschitz stability estimates can be derived under $C^{1,\alpha}$ regularity assumptions on the interfaces.

math.AP↗

arXiv2013

Stable determination of an inclusion in an elastic body by boundary measurements (unabridged)

We consider the inverse problem of identifying an unknown inclusion contained in an elastic body by the Dirichlet-to-Neumann map. The body is made by linearly elastic, homogeneous and isotropic material. The Lam\'e moduli of the inclusion are constant and different from those of the surrounding material. Under mild a-priori regularity assumptions on the unknown defect, we establish a logarithmic stability estimate. For the proof, we extend the approach used for electrical and thermal conductors in a novel way. Main tools are propagation of smallness arguments based on three-spheres inequality for solutions to the Lam\'e system and refined local approximation of the fundamental solution of the Lam\'e system in presence of an inclusion.

math.AP↗

arXiv2012

Recent results about the detection of unknown boundaries and inclusions in elastic plates

In this paper we review some recent results concerning inverse problems for thin elastic plates. The plate is assumed to be made by non-homogeneous linearly elastic material belonging to a general class of anisotropy. A first group of results concerns uniqueness and stability for the determination of unknown boundaries, including the cases of cavities and rigid inclusions. In the second group of results, we consider upper and lower estimates of the area of unknown inclusions given in terms of the work exerted by a couple field applied at the boundary of the plate. In particular, we extend previous size estimates for elastic inclusions to the case of cavities and rigid inclusions. Key words: inverse problems, elastic plates, uniqueness, stability estimates, size estimates, three sphere inequality, unique continuation.

math.AP↗