arXiv subjects

Nadia Ansini

Publications and source records attributed to Nadia Ansini.

6 recordsLinked to original sources

arXiv2026

Multiscale analysis and homogenization of nonlocal thin films

In this paper, we introduce a nonlocal, variational model for thin films. We consider convolution-type functionals defined on a thin domain whose thickness is of order $\gamma$, where the effective interactions range between points is of order $\varepsilon$. We study the $\Gamma$-convergence of these energies, as both parameters vanish, to a local integral functional defined on a lower-dimensional domain. In the periodic homogenization setting, the limit energy density is characterized by an asymptotic formula that depends on the interplay between $\varepsilon$ and $\gamma$. Under suitable assumptions, this formula exhibits a separation of scales effect, namely, the limit energy can be obtained by performing two successive $\Gamma$-limits, first letting one parameter tend to zero while keeping the other fixed.

math.AP↗

arXiv2020

A variational theory of convolution-type functionals

We provide a general treatment of perturbations of a class of functionals modeled on convolution energies with integrable kernel which approximate the $p$-th norm of the gradient as the kernel is scaled by letting a small parameter $\varepsilon$ tend to $0$. We first provide the necessary functional-analytic tools to show coerciveness in $L^p$. The main result is a compactness and integral-representation theorem which shows that limits of convolution-type energies is a standard local integral functional with $p$-growth defined on a Sobolev space. This result is applied to obtain periodic homogenization results, to study applications to functionals defined on point-clouds, to stochastic homogenization and to the study of limits of the related gradient flows.

math.AP↗

arXiv2016

Minimising movements for oscillating energies: the critical regime

Minimising movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimising movement scheme involves a temporal parameter $τ$ and a spatial parameter $ε$, with $τ$ describing the time step and the frequency of the oscillations being proportional to $\frac 1 ε$. The extreme cases of fast time scales $τ<< ε$ and slow time scales $ε<< τ$ have been investigated in Braides, Springer Lecture Notes 2094 (2014). In this article, the intermediate (critical) case of finite ratio $ε/τ>0$ is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterisation of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenised motion are determined.

math.AP↗

arXiv2006

The Neumann sieve problem and dimensional reduction: a multiscale approach

We perform a multiscale analysis for the elastic energy of a $n$-dimensional bilayer thin film of thickness $2δ$ whose layers are connected through an $ε$-periodically distributed contact zone. Describing the contact zone as a union of $(n-1)$-dimensional balls of radius $r\ll ε$ (the holes of the sieve) and assuming that $δ\ll ε$, we show that the asymptotic memory of the sieve (as $ε\to 0$) is witnessed by the presence of an extra interfacial energy term. Moreover we find three different limit behaviors (or regimes) depending on the mutual vanishing rate of $δ$ and $r$. We also give an explicit nonlinear capacitary-type formula for the interfacial energy density in each regime.

math.AP↗

arXiv2000

Homogenization of oscillating boundaries and applications to thin films

We prove a homogenization result for integral functionals in domains with oscillating boundaries, showing that the limit is defined on a degenerate Sobolev space. We apply this result to the description of the asymptotic behaviour of thin films with fast-oscillating profile, proving that they can be described by first applying the homogenization result above and subsequently a dimension-reduction technique.

math.FA↗

arXiv2000

Homogenization of Periodic Multi-Dimensional Structures II: The Linearly Elastic/Perfectly Plastic Case

In this paper we study the asymptotic behaviour via Gamma-convergence of some integral functionals which model some multi-dimensional structures and depend explicitly on the linearized strain tensor. The functionals are defined in particular classes of functions with bounded deformation while the limit problem is set in the usual framework of Sobolev spaces or BD. We also construct an example of such functionals showing that under some special assumptions we can have non local effects.

math.FA↗