Research activity documented by one monograph, one book chapter, and twelve peer-reviewed articles published in international journals.
The complete publication list is reported below, together with selected ongoing projects.
Whenever possible, Published versions are provided; otherwise, Preprint versions are made available.
J. Duran, A. Maione - Editors (Book)
New Frontiers in Homogenization and Fractional Calculus
Trends in Mathematics series, Springer-Birkhäuser
This volume is based on the talks presented at the "New Frontiers in Homogenization and Fractional Calculus" conference, held at CRM-Barcelona on March 24–25, 2025. This event was organized to celebrate the 50th anniversary of the mathematical technique of Gamma-convergence, introduced by Ennio De Giorgi and Tullio Franzoni in 1975. Originally developed for applied purposes, this technique remains a fundamental tool in the study of various scientific phenomena, such as homogenization, phase transitions, and the asymptotic analysis of partial differential equations. At the same time, there has been a growing interest in the study of nonlocal problems, particularly due to their relevance in probability theory. In addition to representing a particularly challenging area of mathematics, these problems have become increasingly relevant in material science applications.
The contributions in this volume cover a wide range of topics and reflect the main areas of current research in variational convergence and nonlocal analysis. Additionally, as these techniques extend beyond pure analytical theory, the contributions explore numerical applications. This volume is aimed at students and researchers in mathematics, physics, and engineering who are interested in exploring recent developments in the theories of homogenization and fractional calculus.
A. Maione
Fractional Sobolev spaces via interpolation, and applications to mixed local-nonlocal operators
To appear in the contributed volume:
"New Frontiers in Homogenization and Fractional Calculus" Trends in Mathematics series, Springer-Birkhäuser (2026)
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.
M. Caponi, A. Carbotti, A. Maione
H-compactness for nonlocal linear operators in fractional divergence form
Calculus of Variations and Partial Differential Equations (2025)
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 Γ-convergence of the associated energies
P. Dondl, A. Maione, S. Wolff-Vorbeck
Phase Field Model for Multi-Material Shape Optimization of Inextensible Rods
ESAIM: Control, Optimisation and Calculus of Variations (2024)
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.
N. Cangiotti, M. Caponi, A. Maione, E. Vitillaro
Schrödinger-Maxwell equations driven by mixed local-nonlocal operators
Fractional Calculus and Applied Analysis (2024)
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
N. Cangiotti, M. Caponi, A. Maione, E. Vitillaro
Klein-Gordon-Maxwell equations driven by mixed local-nonlocal operators
Milan Journal of Mathematics (2023)
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.
A. Maione, D. Mugnai, E. Vecchi
Variational methods for nonpositive mixed local-nonlocal operators
Fractional Calculus and Applied Analysis (2023)
We prove the existence of a weak solution for boundary value problems driven by a mixed local-nonlocal operator.
The main novelty is that such an operator is allowed to be nonpositive definite.
A. Maione, F. Paronetto, E. Vecchi
G-convergence of elliptic and parabolic operators depending on vector fields
ESAIM: Control, Optimisation and Calculus of Variations (2023)
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.
A. Maione, A. Pinamonti, F. Serra Cassano
Γ-Convergence for Functionals Depending on Vector Fields. II. Convergence of Minimizers
SIAM Journal on Mathematical Analysis (2022)
Given a family of locally Lipschitz vector fields X(x)=(X_1(x),...,X_m(x))$ on R^n, m\leq n, we study integral functionals depending on X.
Using the results in [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
A. Maione, A.M. Salort, E. Vecchi
Maz'ya-Shaposhnikova formula in Magnetic Fractional Orlicz-Sobolev spaces
Asymptotic Analysis (2022)
In this note we prove the validity of the Maz'ya-Shaposhnikova formula in magnetic fractional Orlicz-Sobolev spaces.
This complements a previous study of the limit as s\uparrow 1 performed by the second author in [BS2]
A. Maione
H-convergence for equations depending on monotone operators in Carnot groups
Electronic Journal of Differential Equations (2021)
This article presents some results related to the convergence of solutions and momenta of Dirichlet problems for sequences of monotone operators in the sub-Riemannian framework of Carnot groups
M. Capolli, A. Maione, A.M. Salort, E. Vecchi
Asymptotic behaviours in Fractional Orlicz-Sobolev spaces on Carnot groups
The Journal of Geometric Analysis (2021)
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
A. Maione, A. Pinamonti, F. Serra Cassano
Γ-convergence for functionals depending on vector fields. I. Integral representation and compactness
Journal de Mathématiques Pures et Appliquées. Neuvième Série (2020)
Given a family of locally Lipschitz vector fields X(x)=(X_1(x),\dots,X_m(x)) on Rn, m\leq n, we study functionals depending on X.
We prove an integral representation for local functionals with respect to X and a result of Γ-compactness for a class of integral functionals depending on X
A. Maione, E. Vecchi
Integral representation of local left-invariant functionals in Carnot groups
Analysis and Geometry in Metric Spaces (2020)
The aim of this note is to prove a representation theorem for left-invariant functionals in Carnot groups.
As a direct consequence, we can also provide a Γ-convergence result for a smaller class of functionals.
A. Maione, F. Paronetto, S. Verzellesi
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, Poincaré inequalities and compactness properties.
We prove several Γ-convergence results, and apply the latter to the study of H-convergence of anisotropic linear differential operators.
A. Maione (Book)
Lectures on Γ-convergence: Introduction to Theory, Applications, and Emerging Directions in its 50th Year,
Springer-Birkhäuser
G.C. Brusca, M. Caponi, A. Carbotti, A. Maione, F. Paronetto
H-compactness for nonlocal monotone operators in fractional divergence form