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Sergio Vessella

Publications and source records attributed to Sergio Vessella.

At least 19 recordsLinked to original sources
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.

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arXiv2025

Doubling Inequality and Strong Unique Continuation for an Elliptic Transmission Problem

We investigate the Strong Unique Continuation Property (SUCP) for elliptic equations with piecewise Lipschitz coefficients exhibiting jump discontinuities across a regular interface. We prove SUCP at the interface using a doubling inequality derived from a Carleman estimate with a singular weight. This result is intended as a first step toward solving the inverse problem of estimating the size of an unknown, merely measurable, inclusion inside a conductor from boundary measurements.

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arXiv2024

On an inverse problem with applications in cardiac electrophysiology

In this paper, we consider the monodomain model of cardiac electrophysiology. After an analysis of the well-posedness of the forward problem, we show that perfectly insulating regions (modeling ischemic regions in the cardiac tissue) can be uniquely determined by partial boundary measurements of the potential.

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arXiv2023

Notes on unique continuation properties for Partial Differential Equations -- Introduction to the stability estimates for inverse problems

These Notes are intended for graduate or undergraduate students who have familiarity with Lebesgue measure theory, partial differential equations, and functional analysis. The main topics covered in this work are the study of the Cauchy problem and unique continuation properties associated with partial differential equations. The primary objective is to familiarize students with stability estimates in inverse problems and quantitative estimates of unique continuation. The treatment is presented in a self-contained manner.

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arXiv2023

Early-warning inverse source problem for the elasto-gravitational equations

Through coupled physics, we study an early-warning inverse source problem for the elasto-gravitational equations. It consists of a mixed hyperbolic-elliptic system of partial differential equations describing elastic wave displacement and gravity perturbations produced by a source in a homogeneous bounded medium. Within the Cowling approximation, we prove uniqueness and Lipschitz stability for the inverse problem of recovering the moment tensor and the location of the source from early-time measurements of the changes of the gravitational field. The setup studied in this paper is motivated by gravity-based earthquake early warning systems, which are gaining much attention recently.

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arXiv2023

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.

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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.

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arXiv2022

Lipschitz stable determination of polyhedral conductivity inclusions from local boundary measurements

We consider the problem of determining a polyhedral conductivity inclusion embedded in a homogeneous isotropic medium from boundary measurements. We prove global Lipschitz stability for the polyhedral inclusion from the local Dirichlet-to-Neumann map extending in a highly nontrivial way the results obtained in [18] and [19] in the two-dimensional case to the three-dimensional setting.

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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).

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arXiv2020

Lipschitz stable determination of polygonal conductivity inclusions in a layered medium from the Dirichlet to Neumann map

Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann map with respect to movements of a polygonal conductivity inclusion, [11], we extend the results obtained in [8] proving global Lipschitz stability for the determination of a polygonal conductivity inclusion embedded in a layered medium from knowledge of the Dirichlet to Neumann map.

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arXiv2020

Propagation of smallness and size estimate in the second order elliptic equation with discontinuous complex Lipschitz conductivity

In this paper, we would like to derive three-ball inequalities and propagation of smallness for the complex second order elliptic equation with discontinuous Lipschitz coefficients. As an application of such estimates, we study the size estimate problem by one pair of Cauchy data on the boundary. The main ingredient in the derivation of three-ball inequalities and propagation of smallness is a local Carleman proved in our recent paper [FVW].

math.AP↗
arXiv2020

Existence and regularity for eddy currents system with non-smooth conductivity

We discuss the well-posedness of the 'transient eddy current' magneto-quasistatic approximation of Maxwell's initial value problem with bounded and measurable conductivity, with sources, on a domain. We prove existence and uniqueness of weak solutions, and we provide global Hoelder estimates for the magnetic part.

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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}.

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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.

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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.

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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.

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arXiv2018

A transmission problem on a polygonal partition: regularity and shape differentiability

We consider a transmission problem on a polygonal partition for the two-dimensional conductivity equation. For suitable classes of partitions we establish the exact behaviour of the gradient of solutions in a neighbourhood of the vertexes of the partition. This allows to prove shape differentiability of solutions and to establish an explicit formula for the shape derivative.

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