arXiv subjects

Giuseppe Cosma Brusca

Publications and source records attributed to Giuseppe Cosma Brusca.

9 recordsLinked to original sources

arXiv2026

Emergence of a convex strange term via homogenization of non-local energies at the critical exponent

We derive the $Γ$-limit of convolution-type and discrete energies subject to Dirichlet boundary conditions on periodically perforated domains at the critical exponent. We assume that the length-scale of the non-local interactions is much smaller than the side-length of the cubic perforations and prove that a separation of scales occurs. Exploiting the analogies between the variational frameworks of our interest, we employ a unified argument to overcome the technical difficulties that arise from the scaling invariance of the energies. Our multiscale analysis yields a novel observation that is not related to the non-local nature of the functionals, but rather to the analysis at the critical exponent: we prove that the energy density of the $\textit{strange term}$ is convex, even in the vector-valued setting.

math.AP↗

arXiv2026

$Γ$-convergence of convolution-type functionals for free discontinuity problems

We prove compactness with respect to $Γ$-convergence for a general class of non-local energies modelled after the ones considered in [Gobbino, CPAM (1998)]. We give an integral representation result for the limits, which are free discontinuity functionals defined on the space of generalised special functions of bounded variation. We then characterise the bulk and surface energy densities of the obtained limits by means of minimisation problems on small cubes for the approximating energies.

math.AP↗

arXiv2025

Multiscale homogenization of non-local energies of convolution-type

We analyze a family of non-local integral functionals of convolution-type depending on two small positive parameters $\varepsilon,δ$: the first rules the length-scale of the non-local interactions and produces a `localization' effect as it tends to $0$, the second is the scale of oscillation of a finely inhomogeneous periodic structure in the domain. We prove that a separation of the two scales occurs and that the interplay between the localization and homogenization effects in the asymptotic analysis is determined by the parameter $λ$ defined as the limit of the ratio $\varepsilon/δ$. We compute the $Γ$-limit of the functionals with respect to the strong $L^p$-topology for each possible value of $λ$ and detect three different regimes, the critical scale being obtained when $λ\in(0,+\infty)$.

math.AP↗

arXiv2025

Singular perturbations models in phase transitions for anisotropic higher-order materials

We discuss a model for phase transitions in which a double-well potential is singularly perturbed by possibly several terms involving different, arbitrarily high orders of derivation. We study by $Γ$-convergence the asymptotic behaviour as $\varepsilon\to 0$ of the functionals \begin{equation*} F_\varepsilon(u):=\int_Ω\Bigl[\frac{1}{\varepsilon}W(u)+\sum_{\ell=1}^{k}q_\ell\varepsilon^{2\ell-1}|\nabla^{(\ell)}u|_\ell^2\Bigr]\,dx, \qquad u\in H^k(Ω), \end{equation*} for fixed $k>1$ integer, addressing also to the case in which the coefficients $q_1,...,q_{k-1}$ are negative and $|\cdot|_\ell$ is any norm on the space of symmetric $\ell$-tensors for each $\ell\in\{1,...,k\}$. The negativity of the coefficients leads to the lack of a priori bounds on the functionals; such issue is overcome by proving a nonlinear interpolation inequality. With this inequality at our disposal, a compactness result is achieved by resorting to the recent paper [10]. A further difficulty is the presence of general tensor norms which carry anisotropies, making standard slicing arguments not suitable. We prove that the $Γ$-limit is finite only on sharp interfaces and that it equals an anisotropic perimeter, with a surface energy density described by a cell formula.

math.AP↗

arXiv2025

A note on non-local Sobolev spaces and non-local perimeters

We investigate the space of non-local Sobolev functions associated with an integral kernel. We prove an extension result, Sobolev and Poincaré inequalities and an isoperimetric inequality for the non-local perimeter restricted to a set. Finally, we remark on non-local isoperimetric problems, even when the underlying kernel is not necessarily radially symmetric.

math.FA↗

arXiv2024

Higher-order singular perturbation models for phase transitions

Variational models of phase transitions take into account double-well energies singularly perturbed by gradient terms, such as the Cahn-Hilliard free energy. The derivation by $Γ$-convergence of a sharp-interface limit for such energy is a classical result by Modica and Mortola. We consider a singular perturbation of a double-well energy by derivatives of order $k$, and show that we still can describe the limit as in the case $k=1$ with a suitable interfacial energy density, in accord with the case $k=1$ and with the case $k=2$ previously analyzed by Fonseca and Mantegazza. The main isssue is the derivation of an optimal-profile problem on the real line describing the interfacial energy density, which must be conveniently approximated by minimum problems on finite intervals with homogeneous condition on the derivatives at the endpoints up to order $k-1$. To that end a careful study must be carried on of sets where sequences of functions with equibounded energy are ``close to the wells'' and have ``small derivatives'', in terms of interpolation inequalities and energy estimates.

math.AP↗

arXiv2023

Another look at elliptic homogenization

We consider the limit of sequences of normalized $(s,2)$-Gagliardo seminorms with an oscillating coefficient as $s\to 1$. In a seminal paper by Bourgain, Brezis and Mironescu (subsequently extended by Ponce) it is proven that if the coefficient is constant then this sequence $Γ$-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by $\varepsilon$ the scale of the oscillations and we assume that $1-s<\!<\varepsilon^2$, this sequence converges to the homogenized functional formally obtained by separating the effects of $s$ and $\varepsilon$; that is, by the homogenization as $\varepsilon\to 0$ of the Dirichlet integral with oscillating coefficient obtained by formally letting $s\to 1$ first.

math.AP↗

arXiv2023

Homogenization in perforated domains at the critical scale

We describe the asymptotic behaviour of the minimal heterogeneous $d$-capacity of a small set, which we assume to be a ball for simplicity, in a fixed bounded open set $Ω\subseteq \mathbb{R}^d$, with $d\geq2$. Two parameters are involved: $\varepsilon$, the radius of the ball, and $δ$, the length scale of the heterogeneity of the medium. We prove that this capacity behaves as $C|\log \varepsilon|^{d-1}$, where $C=C(λ)$ is an explicit constant depending on the parameter $λ:=\lim_{\varepsilon\to0}|\log δ|/|\log\varepsilon|$. Applying this result, we determine the $Γ$-limit of oscillating integral functionals subjected to Dirichlet boundary conditions on periodically perforated domains. In this instance, our first result is used to study the behaviour of the functionals near the perforations which are exactly balls of radius $\varepsilon$. We prove that, as in the homogeneous case, these lead to an additional term that involves $C(λ)$.

math.AP↗

arXiv2022

Asymptotic behaviour of the capacity in two-dimensional heterogeneous media

We describe the asymptotic behaviour of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set $Ω$. This problem is governed by two small parameters: $\varepsilon$, the size of the inclusion (which is not restrictive to assume to be a ball), and $δ$, the period of the inhomogeneity modelled by oscillating coefficients. We show that this capacity behaves as $C|\log\e|^{-1}$. The coefficient $C$ is explicitly computed from the minimum of the oscillating coefficient and the determinant of the corresponding homogenized matrix, through a harmonic mean with a proportion depending on the asymptotic behaviour of $|\logδ|/|\log\varepsilon|$.

math.AP↗