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Caterina Ida Zeppieri

Publications and source records attributed to Caterina Ida Zeppieri.

16 recordsLinked to original sources
arXiv2026

Homogenisation of phase-field functionals with linear growth

We propose a first rigorous homogenisation procedure in image-segmentation models by analysing the relative impact of (possibly random) fine-scale oscillations and phase-field regularisations for a family of elliptic functionals of Ambrosio and Tortorelli type, when the regularised volume term grows \emph{linearly} in the gradient variable. In contrast to the more classical case of superlinear growth, we show that our functionals homogenise to a free-discontinuity energy whose surface term explicitly depends on the jump amplitude of the limit variable. The convergence result as above is obtained under very mild assumptions which allow us to treat, among other, the case of \emph{stationary random integrands}.

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arXiv2026

Stochastic homogenization of fractional obstacle problems

We prove a stochastic homogenization result for a class of \emph{nonlinear} and \emph{nonlocal} variational problems in domains with many small randomly distributed (bilateral) obstacles. Our model case is a Dirichlet problem for the \emph{fractional} $p$-Laplacian, $p>1$, where a pinning condition $u=0$ is imposed on the solution in a \emph{random} collection of small balls whose centers and radii are generated by a \emph{stationary marked point process}. Such a general obstacle distribution allows for \emph{clustering effects} to appear with positive probability. Under suitable moment conditions on the obstacle radii, we identify a critical scaling regime in which the fractional $p$-capacity density of the obstacles is asymptotically additive \emph{almost surely}. In turn, this key property allows us to derive an effective homogenized problem which is formally analogous to the one obtained in the periodic setting or under the assumption of well-separation for the obstacles. The analysis also extends to the case of \emph{randomly shaped obstacles} and to a broad class of \emph{nonlocal interaction kernels}. At the methodological level, the paper develops a streamlined proof strategy with several new ingredients, among them the use of Palm measures.

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arXiv2023

Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations

In this paper we study the convergence of integral functionals with $q$-growth in a randomly perforated domain of $\mathbb R^n$, with $1<q<n$. Under the assumption that the perforations are small balls whose centres and radii are generated by a \emph{stationary short-range marked point process}, we obtain in the critical-scaling limit an averaged analogue of the nonlinear capacitary term obtained by Ansini and Braides in the deterministic periodic case \cite{Ansini-Braides}. In analogy to the random setting introduced by Giunti, Höfer, and Velázquez \cite{Giunti-Hofer-Velasquez} to study the Poisson equation, we only require that the random radii have finite $(n-q)$-moment. This assumption on the one hand ensures that the expectation of the nonlinear $q$-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.

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arXiv2022

Stochastic homogenization of degenerate integral functionals with linear growth

We study the limit behaviour of a sequence of non-convex, vectorial, random integral functionals, defined on $W^{1,1}$, whose integrands satisfy degenerate linear growth conditions. These involve suitable random, scale-dependent weight-functions. Under minimal assumptions on the integrand and on the weight-functions, we show that the sequence of functionals homogenizes to a non-degenerate functional defined on $BV$.

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arXiv2022

Gradient damage models for heterogeneous materials

In this paper we study the asymptotic behaviour of phase-field functionals of Am brosio and Tortorelli type allowing for small-scale oscillations both in the volume and in the diffuse surface term. The functionals under examination can be interpreted as an instance of a static gradient damage model for heterogeneous materials. Depending on the mutual vanishing rate of the approximation and of the oscillation parameters, the effective behaviour of the model is fully characterised by means of Gamma-convergence.

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arXiv2021

$Γ$-convergence and stochastic homogenisation of singularly-perturbed elliptic functionals

We study the limit behaviour of singularly-perturbed elliptic functionals of the form \[ \mathcal F_k(u,v)=\int_A v^2\,f_k(x,\nabla u)dx+\frac{1}{\varepsilon_k}\int_A g_k(x,v,\varepsilon_k\nabla v)dx\,, \] where $u$ is a vector-valued Sobolev function, $v \in [0,1]$ a phase-field variable, and $\varepsilon_k>0$ a singular-perturbation parameter, i.e., $\varepsilon_k \to 0$, as $k\to +\infty$. Under mild assumptions on the integrands $f_k$ and $g_k$, we show that if $f_k$ grows superlinearly in the gradient-variable, then the functionals $\mathcal F_k$ $Γ$-converge (up to subsequences) to a brittle energy-functional, i.e., to a free-discontinuity functional whose surface integrand does not depend on the jump-amplitude of $u$. This result is achieved by providing explicit asymptotic formulas for the bulk and surface integrands which show, in particular, that volume and surface term in $\mathcal F_k$ decouple in the limit. The abstract $Γ$-convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.

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arXiv2021

A global method for deterministic and stochastic homogenisation in $BV$

In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under \emph{linear} growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation.

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arXiv2020

Stochastic homogenisation of free-discontinuity functionals in random perforated domains

In this paper we study the asymptotic behaviour of a family of random free-discontinuity energies $E_\varepsilon$ defined on a randomly perforated domain, as $\varepsilon$ goes to zero. The functionals $E_\varepsilon$ model the energy associated to displacements of porous random materials that can develop cracks. To gain compactness for sequences of displacements with bounded energies, we need to overcome the lack of equi-coerciveness of the functionals. We do so by means of an extension result, under the assumption that the random perforations cannot come too close to one another. The limit energy is then obtained in two steps. As a first step we apply a general result of stochastic convergence of free-discontinuity functionals to a modified, coercive version of $E_\varepsilon$. Then the effective volume and surface energy densities are identified by means of a careful limit procedure.

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arXiv2018

$Γ$-Convergence of Free-discontinuity Problems

We study the $Γ$-convergence of sequences of free-discontinuity functionals depending on vector-valued functions $u$ which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of $u$. We obtain three main results: compactness with respect to $Γ$-convergence, representation of the $Γ$-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.

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arXiv2018

Homogenization of high-contrast Mumford-Shah energies

We prove a homogenization result for Mumford-Shah-type energies associated to a brittle composite material with weak inclusions distributed periodically at a scale ${\varepsilon}>0$. The matrix and the inclusions in the material have the same elastic moduli but very different toughness moduli, with the ratio of the toughness modulus in the matrix and in the inclusions being $1/β_{\varepsilon}$, with $β_{\varepsilon}>0$ small. We show that the high-contrast behaviour of the composite leads to the emergence of interesting effects in the limit: The volume and surface energy densities interact by $Γ$-convergence, and the limit volume energy is not a quadratic form in the critical scaling $β_{\varepsilon} = {\varepsilon}$, unlike the ${\varepsilon}$-energies, and unlike the extremal limit cases.

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arXiv2018

Quantitative analysis of finite-difference approximations of free-discontinuity problems

Motivated by applications to image reconstruction, in this paper we analyse a \emph{finite-difference discretisation} of the Ambrosio-Tortorelli functional. Denoted by $\varepsilon$ the elliptic-approximation parameter and by $δ$ the discretisation step-size, we fully describe the relative impact of $\varepsilon$ and $δ$ in terms of $Γ$-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when $\varepsilon$ and $δ$ are of the same order, the underlying lattice structure affects the $Γ$-limit which turns out to be an anisotropic free-discontinuity functional.

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arXiv2017

Stochastic Homogenisation of Free-Discontinuity Problems

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.

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arXiv2015

A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation

In this note we formulate a sufficient condition for the quasiconvexity at $x \mapsto λx$ of certain functionals $I(u)$ which model the stored-energy of elastic materials subject to a deformation $u$. The materials we consider may cavitate, and so we impose the well-known technical condition (INV), due to Müller and Spector, on admissible deformations. Deformations obey the condition $u(x)= λx$ whenever $x$ belongs to the boundary of the domain initially occupied by the material. In terms of the parameters of the models, our analysis provides an explicit upper bound on those $λ>0$ such that $I(u) \geq I(u_λ)$ for all admissible $u$, where $u_λ$ is the linear map $x \mapsto λx$ applied across the entire domain. This is the quasiconvexity condition referred to above.

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arXiv2009

Asymptotic analysis of a second-order singular perturbation model for phase transitions

We consider the problem of the asymptotic description of a family of energies introduced by Coleman and Mizel in the theory of nonlinear second-order materials depending on an extra parameter k. By proving a new nonlinear interpolation inequality, we show that there exists a positive constant k_0 such that, for k<k_0, these energies Gamma-converge to a sharp interface functional. Moreover, for a special choice of the potential term in the energy, we provide an upper bound on the values of k such that minimizers cannot develop oscillations on some fine scale, thus improving previous estimates by Mizel, Peletier and Troy.

math.FA↗
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.

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