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Alberto Maione

Publications and source records attributed to Alberto Maione.

16 recordsLinked to original sources
arXiv2026

$H$-convergence and $\Gamma$-convergence in the Riesz fractional setting: the nonlinear case

This paper concerns the $H$-convergence of nonlinear nonlocal monotone operators defined through the Riesz fractional gradient and divergence. We show that the $H$-convergence in this nonlocal framework is equivalent to the $H$-convergence of the corresponding local one. As a consequence, we obtain a $H$-compactness result for a suitable class of nonlocal monotone operators. We then study the $\Gamma$-convergence of nonlocal energy functionals associated with the subclass of \emph{conservative} monotone operators, proving that it is equivalent to the $\Gamma$-convergence of the corresponding local energies. A key ingredient is a new uniqueness result for the integral representation of both local and nonlocal functionals. As a by-product, we obtain the $\Gamma$-compactness of the class of nonlocal energies under consideration. Finally, we show the equivalence between the $H$-convergence of nonlocal conservative monotone operators and the $\Gamma$-convergence of the associated energy functionals.

math.AP↗
arXiv2025

Fractional Sobolev spaces via interpolation, and applications to mixed local-nonlocal operators

In this note, we present a well-known connection between the Sobolev-Slobodeckij spaces, also known as Fractional Sobolev spaces, and interpolation theory. We show how Sobolev spaces can be equivalently characterized as real and complex interpolation spaces between Lebesgue spaces and integer-order Sobolev spaces. We also state a spectral theorem for the so-called mixed local-nonlocal operators, and show how interpolation theory leads to its proof. This note is intended for early-career researchers, and aims to provide a concise and accessible introduction to the subject.

math.FA↗
arXiv2025

Variational convergences under moving anisotropies

We study the asymptotic behaviour of sequences of integral functionals depending on moving anisotropies. We introduce and describe the relevant functional setting, establishing uniform Meyers-Serrin type approximations, Poincar\'e inequalities and compactness properties. We prove several $\Gamma$-convergence results, and apply the latter to the study of $H$-convergence of anisotropic linear differential operators.

math.AP↗
arXiv2024

$H$-compactness for nonlocal linear operators in fractional divergence form

We study the $H$-convergence of nonlocal linear operators in fractional divergence form, where the oscillations of the matrices are prescribed outside the reference domain. Our compactness argument bypasses the failure of the classical localisation techniques that mismatch with the nonlocal nature of the operators involved. If symmetry is also assumed, we extend the equivalence between the $H$-convergence of the operators and the $\Gamma$-convergence of the associated energies.

math.AP↗
arXiv2024

Phase field model for multi-material shape optimization of inextensible rods

We derive a model for the optimization of the bending and torsional rigidities of non-homogeneous elastic rods. This is achieved by studying a sharp interface shape optimization problem with perimeter penalization, that treats both rigidities as objectives. We then formulate a phase field approximation of the optimization problem and show the convergence to the aforementioned sharp interface model via $Γ$-convergence. In the final part of this work we numerically approximate minimizers of the phase field problem by using a steepest descent approach and relate the resulting optimal shapes to the development of the morphology of plant stems.

math.OC↗
arXiv2023

Schrödinger-Maxwell equations driven by mixed local-nonlocal operators

In this paper we prove existence of solutions to Schrödinger-Maxwell type systems involving mixed local-nonlocal operators. Two different models are considered: classical Schrödinger-Maxwell equations and Schrödinger-Maxwell equations with a coercive potential, and the main novelty is that the nonlocal part of the operator is allowed to be nonpositive definite according to a real parameter. We then provide a range of parameter values to ensure the existence of solitary standing waves, obtained as Mountain Pass critical points for the associated energy functionals.

math.AP↗
arXiv2023

Klein-Gordon-Maxwell equations driven by mixed local-nonlocal operators

Classical results concerning Klein-Gordon-Maxwell type systems are shortly reviewed and generalized to the setting of mixed local-nonlocal operators, where the nonlocal one is allowed to be nonpositive definite according to a real parameter. In this paper, we provide a range of parameter values to ensure the existence of solitary (standing) waves, obtained as Mountain Pass critical points for the associated energy functionals in two different settings, by considering two different classes of potentials: constant potentials and continuous, bounded from below, and coercive potentials.

math.AP↗
arXiv2022

$G$-convergence of elliptic and parabolic operators depending on vector fields

We consider sequences of elliptic and parabolic operators in divergence form and depending on a family of vector fields. We show compactness results with respect to G-convergence, or H-convergence, by means of the compensated compactness theory, in a setting in which the existence of affine functions is not always guaranteed, due to the nature of the family of vector fields.

math.AP↗
arXiv2021

$Γ$-convergence for functionals depending on vector fields. II. Convergence of minimizers

Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study integral functionals depending on $X$. Using the results in \cite{MPSC1}, we study the convergence of minima, minimizers and momenta of those functionals. Moreover, we apply these results to the periodic homogenization in Carnot groups and to prove a $H$-compactness theorem for linear differential operators of the second order depending on $X$.

math.AP↗
arXiv2020

$H$-convergence for equations depending on monotone operators in Carnot groups

Let $Ω$ be an open and bounded subset of a Carnot Group $\mathbb{G}$ and $2\leq p<\infty$. In this paper we present some results related to the convergence of solutions of Dirichlet problems for sequences of monotone operators. The aim of this paper is to give a generalization of well-known results of Tartar, De Arcangelis-Serra Cassano and Baldi-Franchi-Tchou-Tesi in more general frameworks.

math.AP↗
arXiv2019

Asymptotic behaviours in Fractional Orlicz-Sobolev spaces on Carnot groups

In this article we define a class of fractional Orlicz-Sobolev spaces on Carnot groups and, in the spirit of the celebrated results of Bourgain-Brezis-Mironescu and of Maz'ya-Shaposhnikova, we study the asymptotic behavior of the Orlicz functionals when the fractional parameter goes to $1$ and $0$.

math.FA↗