arXiv subjects

Gianluca Orlando

Publications and source records attributed to Gianluca Orlando.

18 recordsLinked to original sources

arXiv2026

Scaling limits for Moran processes on metric strategy spaces: an Eulerian derivation of pure replicator and Fleming--Viot measure-valued PDEs

We study the large-population limit of a discrete-time Moran process featuring multiple strategies, drawn from a possibly infinite strategy space $\mathcal{V}$ (a metric space), under both weak and strong selection. In the Eulerian density formulation, the limiting dynamics depend critically on the relative scaling of population size, mutation rate, and selection intensity. Depending on these scalings, the limit behavior is governed either by a purely deterministic replicator-type continuity equation or by a diffusion-enhanced PDE. In the latter case, the diffusion operator recovers the classical Fleming--Viot operator and the Kimura equation as special instances. Methodologically, we derive the limit by reinterpreting the discrete process in Eulerian coordinates, constructing interpolating curves that satisfy an approximate PDE, and establishing convergence via a compactness argument in a suitable topology on the space of probability measures over probabilities over $\mathcal{V}$.

math.AP↗

arXiv2025

Frequency-dependent damping in the linear wave equation

We propose a model for frequency-dependent damping in the linear wave equation. After proving well-posedness of the problem, we study qualitative properties of the energy. In the one-dimensional case, we provide an explicit analysis for special choices of the damping operator. Finally, we show, in special cases, that solutions split into a dissipative and a conservative part.

math.AP↗

arXiv2025

Replicator dynamics as the large population limit of a discrete Moran process in the weak selection regime: A proof via Eulerian specification

We study the large population limit of a multi-strategy discrete-time Moran process in the weak selection regime. We show that the replicator dynamics is interpreted as the large-population limit of the Moran process. This result is obtained by interpreting the discrete process in its Eulerian specification, proving a compactness result in the Wasserstein space of probability measures for the law of the proportions of strategies, and passing to the limit in the continuity equation that describes the evolution of the proportions.

math.AP↗

arXiv2024

Comparison between solutions to the linear peridynamics model and solutions to the classical wave equation

In this paper, we consider an equation inspired by linear peridynamics and we establish its connection with the classical wave equation. In particular, given a horizon $δ>0$ accounting for the region of influence around a material point, we prove existence and uniqueness of a solution $u_δ$ and demonstrate the convergence of $u_δ$ to solutions to the classical wave equation as $δ\to 0$. Moreover, we prove that the solutions to the peridynamics model with small frequency initial data are close to solutions to the classical wave equation.

math.AP↗

arXiv2024

Stacking faults in the limit of a discrete model for partial edge dislocations

In the limit of vanishing lattice spacing we provide a rigorous variational coarse-graining result for a next-to-nearest neighbor lattice model of a simple crystal. We show that the $Γ$-limit of suitable scaled versions of the model leads to an energy describing a continuum mechanical model depending on partial dislocations and stacking faults. Our result highlights the necessary multiscale character of the energies setting the groundwork for more comprehensive models that can better explain and predict the mechanical behavior of materials with complex defect structures.

math.AP↗

arXiv2023

Exponential convergence to steady-states for trajectories of a damped dynamical system modelling adhesive strings

We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modelling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.

math.AP↗

arXiv2023

Mean-field optimal control in a multi-agent interaction model for prevention of maritime crime

We study a multi-agent system for the modeling maritime crime. The model involves three interacting populations of ships: commercial ships, pirate ships, and coast guard ships. Commercial ships follow commercial routes, are subject to traffic congestion, and are repelled by pirate ships. Pirate ships travel stochastically, are attracted by commercial ships and repelled by coast guard ships. Coast guard ships are controlled. We prove well-posedness of the model and existence of optimal controls that minimize dangerous contacts. Then we study, in a two-step procedure, the mean-field limit as the number of commercial ships and pirate ships is large, deriving a mean-field PDE/PDE/ODE model. Via $Γ$-convergence, we study the limit of the corresponding optimal control problems.

math.OC↗

arXiv2021

Variational analysis of the $J_1$-$J_2$-$J_3$ model: a non-linear lattice version of the Aviles-Giga functional

We study the variational limit of the frustrated $J_1$-$J_2$-$J_3$ spin model on the square lattice in the vicinity of the ferromagnet/helimagnet transition point as the lattice spacing vanishes. We carry out the $Γ$-convergence analysis of proper scalings of the energy and we characterize the optimal cost of a chirality transition in $BV$ proving that the system is asymptotically driven by a discrete version of a non-linear perturbation of the Aviles-Giga energy functional.

math.AP↗

arXiv2021

Emergence of concentration effects in the variational analysis of the $N$-clock model

We investigate the relationship between the $N$-clock model (also known as planar Potts model or $\mathbb{Z}_N$-model) and the $XY$ model (at zero temperature) through a $Γ$-convergence analysis of a suitable rescaling of the energy as both the number of particles and $N$ diverge. We prove the existence of rates of divergence of $N$ for which the continuum limits of the two models differ. With the aid of Cartesian currents we show that the asymptotics of the $N$-clock model in this regime features an energy which may concentrate on geometric objects of various dimensions. This energy prevails over the usual vortex-vortex interaction energy.

math-ph↗

arXiv2020

The $N$-clock model: Variational analysis for fast and slow divergence rates of $N$

We study a nearest neighbors ferromagnetic spin system on the square lattice in which the spin field is constrained to take values in a discretization of the unit circle consisting of $N$ equi-spaced vectors, also known as $N$-clock model. We find a fast rate of divergence of $N$ with respect to the lattice spacing for which the $N$-clock model has the same discrete-to-continuum variational limit of the $XY$ model, in particular concentrating energy on topological defects of dimension 0. We prove the existence of a slow rate of divergence of $N$ at which the coarse-grain limit does not detect topological defects, but it is instead a $BV$-total variation. Finally, the two different types of limit behaviors are coupled in a critical regime for $N$, whose analysis requires the aid of Cartesian currents.

math-ph↗

arXiv2020

The antiferromagnetic XY model on the triangular lattice: topological singularities

We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice in the vortex regime. Within this regime, the spin system cannot overcome the energetic barrier of chirality transitions, hence one of the two chirality phases is prevalent. We find the order parameter that describes the vortex structure of the spin field in the majority chirality phase and we compute explicitly the $Γ$-limit of the scaled energy, showing that it concentrates on finitely many vortex-like singularities of the spin field.

math-ph↗

arXiv2020

Fatigue effects in elastic materials with variational damage models: A vanishing viscosity approach

We study the existence of quasistatic evolutions for a family of gradient damage models which take into account fatigue, that is the process of weakening in a material due to repeated applied loads. The main feature of these models is the fact that damage is favoured in regions where the cumulation of the elastic strain (or other relevant variables, depend on the model) is higher. To prove the existence of a quasistatic evolution, we follow a vanishing viscosity approach based on two steps: we first let the time-step $τ$ of the time-discretisation and later the viscosity parameter $ε$ go to zero. As $τ\to 0$, we find $ε$-approximate viscous evolutions; then, as $ε\to 0$, we find a rescaled approximate evolution satisfying an energy-dissipation balance.

math.AP↗

arXiv2020

Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model

We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $\varepsilon$-lattice in which the spin field is constrained to take values in a discretization $\mathcal{S}_N$ of the unit circle~$\mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $Γ$-convergence analysis consists of two steps: we first fix $N$ and let the lattice spacing $\varepsilon \to 0$, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in $\mathcal{S}_N$; at a second stage, we let $N \to +\infty$. The final result of this two-step limit process is an anisotropic total variation of $\mathbb{S}^1$-valued vector fields of bounded variation.

math-ph↗

arXiv2020

The antiferromagnetic XY model on the triangular lattice: chirality transitions at the surface scaling

We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice. The system is fully frustrated and displays two families of ground states distinguished by the chirality of the spin field. We compute the Γ-limit of the energy in a regime which detects chirality transitions on one-dimensional interfaces between the two admissible chirality phases.

math.AP↗

arXiv2019

A lower semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,γ}$, $γ>1$

We prove the lower semicontinuity of functionals of the form \[ \int \limits_Ω\! V(α) \, \mathrm{d} |\mathrm{E} u| \, , \] with respect to the weak converge of $α$ in $W^{1,γ}(Ω)$, $γ> 1$, and the weak* convergence of $u$ in $BD(Ω)$, where $Ω\subset \mathbb{R}^n$. These functional arise in the variational modelling of linearised elasto-plasticity coupled with damage and their lower semicontinuity is crucial in the proof of existence of quasi-static evolutions. This is the first result achieved for subcritical exponents $γ< n$.

math.AP↗

arXiv2019

Variational analysis of a two-dimensional frustrated spin system: emergence and rigidity of chirality transitions

We study the discrete-to-continuum variational limit of the $J_{1}$-$J_{3}$ spin model on the square lattice in the vicinity of the helimagnet/ferromagnet transition point as the lattice spacing vanishes. Carrying out the $Γ$-convergence analysis of proper scalings of the energy, we prove the emergence and characterize the geometric rigidity of the chirality phase transitions.

math.AP↗

arXiv2018

Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue

In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e., a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement.

math.AP↗

arXiv2017

A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,n}$

In this paper we prove a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures $α_k\,\mathrm{E}u_k$ with respect to the weak convergence $α_k\rightharpoonup α$ in $W^{1,n}(Ω)$ and the weak$^*$ convergence $u_k\stackrel{*}\rightharpoonup u$ in $BD(Ω)$, $\mathrm{E}$ denoting the symmetrised gradient. A concentration compactness argument shows that the limit has the form $α\,\mathrm{E}u+η$, with $η$ supported on an at most countable set.

math.AP↗