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Maicol Caponi

Publications and source records attributed to Maicol Caponi.

14 recordsLinked to original sources
arXiv2026

$H$-convergence and $Γ$-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 $Γ$-convergence of nonlocal energy functionals associated with the subclass of \emph{conservative} monotone operators, proving that it is equivalent to the $Γ$-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 $Γ$-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 $Γ$-convergence of the associated energy functionals.

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arXiv2025

A Survey on the Div-Curl Lemma and Some Extensions to Fractional Sobolev Spaces

This survey is a chapter of a forthcoming book. This chapter recalls the classical formulation of the Div-Curl lemma along with its proof, and presents some possible generalizations in the fractional setting, within the framework of the Riesz fractional gradient and divergence introduced by Shieh and Spector (2015) and further developed by Comi and Stefani (2019).

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arXiv2025

$H$-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 $Γ$-convergence of the associated energies.

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arXiv2024

A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations

We derive a strain-gradient theory for plasticity as the $Γ$-limit of discrete dislocation fractional energies, without the introduction of a core-radius. By using the finite horizon fractional gradient introduced by Bellido, Cueto, and Mora-Corral of 2023, we consider a nonlocal model of semi-discrete dislocations, in which the stored elastic energy is computed via the fractional gradient of order $1-α$. As $α$ goes to $0$, we show that suitably rescaled energies $Γ$-converge to the macroscopic strain-gradient model of Garroni, Leoni, and Ponsiglione of 2010.

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arXiv2023

A rate-independent model for geomaterials under compression coupling strain gradient plasticity and damage

We study a strain gradient-enhanced version of a model for geomaterials under compression by Marigo and Kazymyrenko (2019) coupling damage and small-strain associative plasticity. We prove that the jumps in time of the plastic variable may happen only along jumps of the damage variable. Moreover, we perform a vanishing-viscosity analysis showing existence of Balanced Viscosity quasistatic solutions à la Mielke-Rossi-Savaré.

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arXiv2023

Geometric rigidity on Sobolev spaces with variable exponent and applications

We present extensions of rigidity estimates and of Korn's inequality to the setting of (mixed) variable exponents growth. The proof techniques, based on a classical covering argument, rely on the log-Hölder continuity of the exponent to get uniform regularity estimates on each cell of the cover, and on an extension result à la Nitsche in Sobolev spaces with variable exponents. As an application, by means of $Γ$-convergence we perform a passage from nonlinear to linearized elasticity under variable subquadratic energy growth far from the energy well.

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arXiv2023

The viscoelastic paradox in a nonlinear Kelvin-Voigt type model of dynamic fracture

In this paper we consider a dynamic model of fracture for viscoelastic materials, in which the constitutive relation, involving the Cauchy stress and the strain tensors, is given in an implicit nonlinear form. We prove the existence of a solution to the associated viscoelastic dynamic system on a prescribed time-dependent cracked domain via a discretisation-in-time argument. Moreover, we show that such a solution satisfies an energy-dissipation balance in which the energy used to increase the crack does not appear. As a consequence, in analogy to the linear case this nonlinear model exhibits the so-called viscoelastic paradox.

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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.

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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.

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arXiv2020

An existence result for the fractional Kelvin-Voigt's model on time-dependent cracked domains

We prove an existence result for the fractional Kelvin-Voigt's model involving Caputo's derivative on time-dependent cracked domains. We first show the existence of a solution to a regularized version of this problem. Then, we use a compactness argument to derive that the fractional Kelvin-Voigt's model admits a solution which satisfies an energy-dissipation inequality. Finally, we prove that when the crack is not moving, the solution is unique.

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arXiv2020

Existence of solutions to a phase-field model of dynamic fracture with a crack-dependent dissipation

We propose a phase-field model of dynamic fracture based on the Ambrosio--Tortorelli's approximation, which takes into account dissipative effects due to the speed of the crack tips. By adapting the time discretization scheme contained in [C.J. Larsen, C. Ortner, and E. Süli, Math. Models Methods Appl. Sci. (2010)], we show the existence of a dynamic crack evolution satisfying an energy-dissipation balance, according to Griffith's criterion. Finally, we analyze the dynamic phase-field model of [B. Bourdin, C.J. Larsen, and C.L. Richardson, Int. J. Fracture (2011)] and [C.J. Larsen, IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (2010)] with no dissipative terms.

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arXiv2019

A dynamic model for viscoelastic materials with prescribed growing cracks

In this paper we prove the existence of solutions for a class of viscoelastic dynamic systems on time--dependent cracked domains, with possibly degenerate viscosity coefficients. Under stronger regularity assumptions we also show a uniqueness result. Finally, we exhibit an example where the energy--dissipation balance is not satisfied, showing there is an additional dissipation due to the crack growth.

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arXiv2019

Energy-dissipation balance of a smooth moving crack

In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in [S.Nicaise, A.M.Sandig - \textit{J. Math. Anal. Appl.} 2007] valid for straight fractures.

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