Searcharxiv⌕ Search

arXiv subjects

Gianpaolo Piscitelli

Publications and source records attributed to Gianpaolo Piscitelli.

At least 19 recordsLinked to original sources
arXiv2026

Monotonicity Principle and "p-Laplace Signature" for Tomography in Nonlinear Elliptic Inverse Problems

This paper proposes a framework for treating the inverse obstacle problem for nonlinear elliptic equations with nonlinear materials. The problem is challenging because nonlinear materials exhibit a rich diversity of scenarios to consider, since nonlinearity can take different forms. In this article, after categorizing the nonlinearities into a few fundamental classes, a dedicated imaging method is proposed for each class, derived by combining two powerful concepts: the Monotonicity Principle (MP) and the $p-$Laplace Signature (pLS). The Monotonicity Principle (MP), recently extended to nonlinear materials, provides a monotonic relationship between the material property and the measured quantity (the Average Dirichlet-to-Neumann map) that can be \lq\lq inverted\rq\rq \ to find the shape of anomalies. The $p-$Laplace Signature (pLS) allows for modelling the solution of an elliptic PDE with nonlinear materials, for large or small boundary data, in terms of a proper $p-$Laplace equation that captures the essence (the signature) of the problem. For example, pLS with $p=2$ allows the reduction of a nonlinear elliptic PDE to a linear one, providing a powerful bridge for applying imaging methods and algorithms developed for linear materials. In this contribution, the two pillars of MP and pLS are combined in new imaging methods to enlarge the class of nonlinearity that can be treated within the inverse obstacle problem. Moreover, the theoretical limits of the methods are provided in the ideal case of noise-free measurements: outer-support reconstruction when $p=2$, and convex-hull reconstruction when $p\neq2$.

eess.SP↗
arXiv2026

Monotonicity of the Laplace Transform for Tomography in Dissipative Systems

This paper addresses the problem of tomography for the interior of dissipative materials, with a focus on Magnetic Induction Tomography (MIT), a proven technique for imaging the interior of conductive materials using low-frequency electromagnetic fields. Processing MIT data is mathematically challenging because of the non-linear and ill-posed nature of the underlying inverse problem. On the other hand, the Monotonicity Principle is recognized as the basis for developing effective approaches. In this framework, the paper presents a principle of monotonicity for the Transfer Operator in Magnetic Induction Tomography, i.e. the operator mapping the Laplace transform of the applied source onto the Laplace transform of the measured quantity. Specifically, it is proved that the Transfer Operator satisfies a Monotonicity Principle when evaluated on a proper real semi-axis of the complex plane. The description of the related (real-time) imaging method is also given.

math.AP↗
arXiv2025

First Eigenvalue and Torsional Rigidity: Isoperimetric Inequalities for the Fractional Laplacian

We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $Ω\subset \mathbb{R}^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first Dirichlet eigenvalue $λ_1(Ω)$ of the fractional Laplacian attains its minimum on balls. With the same arguments we also establish a reverse Hölder inequality for an eigenfunction corresponding to $λ_1(Ω)$.

math.AP↗
arXiv2025

Symmetry results for a nonlocal nonlinear Poincaré-Wirtinger inequality

In this paper, we study the optimal constant in the nonlocal nonlinear Poincaré-Wirtinger inequality in $(a,b)\subset\mathbb R$: \begin{equation*} λ_α(p,q,r){\left(\int_{a}^{b}|u|^{q}dx\right)^\frac pq}\le{\int_{a}^{b}|u'|^{p}dx+α\left|\int_{a}^{b}|u|^{r-2}u\, dx\right|^{\frac p{r-1}}}, \end{equation*}where $α\in\mathbb R$, $p,q,r >1$ such that $\frac{2p}{p+2}\le q\le p$ and $\frac q2+1\le r \le q+\frac q p$. This problem admits a variational characterization in the nonlocal setting, as the associated Euler-Lagrange equation involves an integral term depending on the unknown function over the entire interval of definition. We prove the existence of a critical value $α_C=α_C (p,q,r)$ such that the minimizers are even and have constant sign for $α\leα_{C}$, while they are odd for $α\geq α_{C}$.

math.AP↗
arXiv2025

On the second anisotropic Cheeger constant and related questions

In this paper we study the behavior of the second eigenfunction of the anisotropic $p$-Laplace operator \[ - Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_ξ(\nabla u)\right), \] as $p \to 1^+$, where $F$ is a suitable smooth norm of $\mathbb R^{n}$. Moreover, for any regular set $Ω$, we define the second anisotropic Cheeger constant as \begin{equation*} h_{2,F}(Ω):=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\},\; E_{1},E_{2}\subset Ω, E_{1}\cap E_{2}=\emptyset\right\}, \end{equation*} where $P_{F}(E)$ is the anisotropic perimeter of $E$, and study the connection with the second eigenvalue of the anisotropic $p$-Laplacian. Finally, we study the twisted anisotropic $q$-Cheeger constant with a volume constraint.

math.AP↗
arXiv2025

The inverse obstacle problem for nonlinear inclusions

The Monotonocity 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 Monotonicity Principle, while results (iii) requires to prove the so-called Converse of the Monotonicity Principle, a theoretical sults of fundamental relevance to evaluate the ideal (noise-free) performances of the imaging method. The results are provided in a quite general setting for Calderòn 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.

math.AP↗
arXiv2024

A sharp bound for the first Robin-Dirichlet eigenvalue

In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell reaches the maximum of the first eigenvalue of this problem among the domains with fixed measure, outer perimeter and inner $(n-1)^{th}$ quermassintegral.

math.AP↗
arXiv2024

A stability result for the first Robin-Neumann eigenvalue: A double perturbation approach

Let $Ω=Ω_0\setminus \overlineΘ\subset \mathbb{R}^n$, $n\geq 2$, where $Ω_0$ and $Θ$ are two open, bounded and convex sets such that $\overlineΘ\subset Ω_0$ and let $β<0$ be a given parameter. We consider the eigenvalue problem for the Laplace operator associated to $Ω$, with Robin boundary condition on $\partial Ω_0$ and Neumann boundary condition on $\partial Θ$. In [Paoli-Piscitelli-Trani, ESAIM-COCV '20] it is proved that the spherical shell is the only maximizer for the first Robin-Neumann eigenvalue in the class of domains $Ω$ with fixed outer perimeter and volume. We establish a quantitative version of the afore-mentioned isoperimetric inequality; the main novelty consists in the introduction of a new type of hybrid asymmetry, that turns out to be the suitable one to treat the different conditions on the outer and internal boundary. Up to our knowledge, in this context, this is the first stability result in which both the outer and the inner boundary are perturbed.

math.AP↗
arXiv2024

Piecewise nonlinear materials and Monotonicity Principle

This paper is focused on the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponent. This results are relevant because enables the use of a fast imaging method based on MP, to the wide class of problems with two or more materials, where at least one is nonlinear. The treatment is very general and allows to model a wide variety of practical configurations such as, for instance, Superconducting (SC) or 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, Inverse Problems 2021], where the MP for a single type of nonlinearity was treated. Realistic numerical examples confirm the theoretical findings.

math.AP↗
arXiv2024

Tomography of nonlinear materials via the Monotonicity Principle

In this paper we present a first non-iterative imaging method for nonlinear materials, based on Monotonicity Principle. Specifically, we deal with the inverse obstacle problem, where the aim is to retrieve a nonlinear anomaly embedded in linear known background. The Monotonicity Principle (MP) is a general property for various class of PDEs, that has recently generalized to nonlinear elliptic PDEs. Basically, it states a monotone relation between the point-wise value of the unknown material property and the boundary measurements. It is at the foundation of a class of non-iterative imaging methods, characterized by a very low execution time that makes them ideal candidates for real-time applications. In this work, we develop an inversion method that overcomes some of the peculiar difficulties in practical application of MP to imaging of nonlinear materials, preserving the feasibility for real-time applications. For the sake of clarity, we focus on a specific application, i.e. the Magnetostatic Permeability Tomography where the goal is retrieving the unknown (nonlinear) permeability by boundary measurements in DC operations. This choice is motivated by applications in the inspection of boxes and containers for security. Reconstructions from simulated data prove the effectiveness of the presented method.

math.NA↗
arXiv2023

Imaging of nonlinear materials via the Monotonicity Principle

Inverse problems, which are related to Maxwell's equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behaviour of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.

math.NA↗
arXiv2023

Symmetrization results for general nonlocal linear ellipitic and parabolic problems

We establish a Talenti-type symmetrization result in the form of mass concentration (i.e. integral comparison) for very general linear nonlocal elliptic problems, equipped with homogeneous Dirichlet boundary conditions. In this framework, the relevant concentration comparison for the classical fractional Laplacian can be reviewed as a special case of our main result, thus generalizing the previous results in [21]. Finally, using an implicit time discretization techniques, similar results are obtained for the solutions of Cauchy-Dirichlet nonlocal linear parabolic problems.

math.AP↗
arXiv2023

On the first Robin eigenvalue of the Finsler $p$-Laplace operator as $p\to 1$

Let $Ω$ be a bounded, connected, sufficiently smooth open set, $p>1$ and $β\in\mathbb R$. In this paper, we study the $Γ$-convergence, as $p\rightarrow 1^+$, of the functional \[ J_p(φ)=\frac{\int_ΩF^p(\nabla φ)dx+β\int_{\partial Ω} |φ|^pF(ν)d\mathcal{H}^{N-1}}{\int_Ω|φ|^pdx} \] where $φ\in W^{1,p}(Ω)\setminus\{0\}$ and $F$ is a sufficientely smooth norm on $\mathbb R^n$. We study the limit of the first eigenvalue $λ_1(Ω,p,β)=\inf_{\substack{φ\in W^{1,p}(Ω)\\ φ\ne 0}}J_p(φ)$, as $p\to 1^+$, that is: \begin{equation*} Λ(Ω,β)=\inf_{\substack{φ\in BV(Ω)\\ φ\not\equiv 0}}\dfrac{|Du|_F(Ω)+\min\{β,1\}\displaystyle \int_{\partial Ω}|φ|F(ν)d\mathcal H^{N-1}}{\displaystyle s\int_Ω|φ|dx}. \end{equation*} Furthermore, for $β>-1$, we obtain an isoperimetric inequality for $Λ(Ω,β)$ depending on $β$. The proof uses an interior approximation result for $BV(Ω)$ functions by $C^\infty(Ω)$ functions in the sense of strict convergence on $\mathbb R^n$ and a trace inequality in $BV$ with respect to the anisotropic total variation.

math.AP↗
arXiv2022

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \begin{equation*} λ_1(β,Ω)=\min_{ψ\in W^{1,p}(Ω)\setminus\{0\} } \frac{\displaystyle\int_ΩF(\nabla ψ)^p dx +β\displaystyle\int_{\partialΩ}|ψ|^pF(ν_Ω) d\mathcal H^{N-1} }{\displaystyle\int_Ω|ψ|^p dx}, \end{equation*} where $p\in]1,+\infty[$, $Ω$ is a bounded, mean convex domain in $\mathbb R^{N}$, $ν_Ω$ is its Euclidean outward normal, $β$ is a real number, and $F$ is a sufficiently smooth norm on $\mathbb R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $β$ and on geometrical quantities associated to $Ω$. More precisely, we prove a lower bound of $λ_{1}$ in the case $β>0$, and a upper bound in the case $β<0$. As a consequence, we prove, for $β>0$, a lower bound for $λ_{1}(β,Ω)$ in terms of the anisotropic inradius of $Ω$ and, for $β<0$, an upper bound of $λ_{1}(β,Ω)$ in terms of $β$.

math.AP↗
arXiv2022

An isoperimetric inequality for the first Steklov-Dirichlet Laplacian eigenvalue of convex sets with a spherical hole

In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if $Ω=Ω_0 \setminus \bar{B}_{R_1}$, where $B_{R_1}$ is the ball centered at the origin with radius $R_1>0$ and $Ω_0\subset\mathbb{R}^n$, $n\geq 2$, is an open bounded and convex set such that $B_{R_1}\Subset Ω_0$, then the first Steklov-Dirichlet eigenvalue $σ_1(Ω)$ has a maximum when $R_1$ and the measure of $Ω$ are fixed. Moreover, if $Ω_0$ is contained in a suitable ball, we prove that the spherical shell is the maximum.

math.AP↗
arXiv2021

The Pseudo-orthogonality for Graph $1$-Laplacian Eigenvectors and Applications to Higher Cheeger Constants and Data Clustering

The data clustering problem consists in dividing a data set into prescribed groups of homogeneous data. This is a NP-hard problem that can be relaxed in the spectral graph theory, where the optimal cuts of a graph are related to the eigenvalues of graph $1$-Laplacian. In this paper, we firstly give new notations to describe the paths, among critical eigenvectors of the graph $1$-Laplacian, realizing sets with prescribed genus. We introduce the pseudo-orthogonality to characterize $m_3(G)$, a special eigenvalue for the graph $1$-Laplacian. Furthermore, we use it to give an upper bound for the third graph Cheeger constant $h_3(G)$, that is $h_3(G) \le m_3(G)$. This is a first step for proving that the $k$-th Cheeger constant is the minimum of the $1$-Laplacian Raylegh quotient among vectors that are pseudo-orthogonal to the vectors realizing the previous $k-1$ Cheeger constants. Eventually, we apply these results to give a method and a numerical algorithm to compute $m_3(G)$, based on a generalized inverse power method.

math.AP↗
arXiv2021

The Monotonicity Principle for Magnetic Induction Tomography

The inverse problem treated in this article consists in reconstructing the electrical conductivity from the free response of the system in the magneto-quasi-stationary (MQS) limit. The MQS limit corresponds to a diffusion PDE. In this framework, a key role is played by the Monotonicity Principle, that is a monotone relation connecting the unknown material property to the (measured) free-response. MP is relevant as basis of noniterative and real-time imaging methods. Monotonicity Principles have been found in many different physical problems governed by PDEs of different nature. Despite its rather general nature, each different physical/mathematical context requires to discover the proper operator showing MP. For doing this, it is necessary to develop ad-hoc mathematical approaches tailored on the specific framework. In this article, we prove a monotonic relationship between the electrical resistivity and the time constants characterizing the free-response for MQS systems. The key result is the representation of the induced current density through a modal representation. The main result is based on the analysis of an elliptic eigenvalue problem, obtained from separation of variables.

math.AP↗