arXiv subjects

Giuliana Fusco

Publications and source records attributed to Giuliana Fusco.

5 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 for nonlocal phase transitions involving the $H^{1/2}$ norm and surfactants

We study functionals \begin{equation*} F_\varepsilon (u,ρ) := \frac{1}{\varepsilon} \int_ΩW(u) \, dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy \,dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\left| \int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy - ρ(x) \right| \,dx \end{equation*} for a double-well potential $W$ and a nonlocal, critically scaled gradient-like term, together with a surfactant term. We show compactness in the space of $BV$ functions on $Ω$ and the $Γ$-convergence to an energy given as local perimeter-type functional, depending also on the limit density of surfactant on the interface, plus the total variation of the surfactant measure away from the interface.

math.AP↗

arXiv2025

Crystalline Motion of discrete interfaces in the Blume-Emery-Griffiths Model: partial wetting

We continue the variational study of the discrete-to-continuum evolution of lattice systems of Blume-Emery-Griffith type which model two immiscible phases in the presence of a surfactant. In our previous work \cite{CFS}, we analyzed the case of a completely wetted crystal and described how the interplay between surfactant evaporation and mass conservation leads to a transition between crystalline mean curvature flow and pinned evolutions. In the present paper, we extend the analysis to the regime of partial wetting, where the surfactant occupies only a portion of the interface. Within the minimizing-movements scheme, we rigorously derive the continuum evolution and show how partial wetting introduces a complex coupling between interfacial motion and redistribution of surfactant. The resulting evolution exhibits new features absent in the fully wetted case, including the coexistence of moving and pinned facets or the emergence and long-lived metastable states. This provides, to our knowledge, the first discrete-to-continuum variational description of partially wetted crystalline interfaces, bridging the gap between microscopic lattice models and experimentally observed surfactant-induced pinning phenomena in immiscible systems.

math.AP↗

arXiv2025

Crystalline motion of discrete interfaces in the Blume-Emery-Griffiths model

We study the discrete-to-continuum evolution of a lattice system consisting of two immiscible phases labelled by -1 and +1 in presence of a surfactant phase labelled by 0. The system's energy is described by the classical Blume-Emery-Griffith model on the lattice epsilon Z^2, and its continuum evolution is obtained as epsilon tends to zero through a minimizing-movements scheme with a time step proportional to epsilon. The dissipation functional we choose contains two contributions: a standard Almgren-Taylor-Wang type term penalizing the distance between successive configurations of the +1 phase, and a term penalizing the variation of the surfactant mass and modeling surfactant evaporation. The latter term depends on a scaling parameter gamma > 0, which determines whether the surfactant mass is conserved at each time step. We focus on the case in which the initial configuration consists of a single crystal of phase 1 completely wetted by the surfactant. For gamma > 2 the surfactant can lose mass and the evolution reduces to the crystalline mean curvature flow of an Ising-type model, while for gamma < 2 the conservation of the surfactant mass leads to a more complex evolution characterized by stronger non-uniqueness and partial pinning.

math.AP↗

arXiv2025

Variational analysis of discrete Dirichlet problems in periodically perforated domains

In this paper we study the asymptotic behavior of a family of discrete functionals as the lattice size, $\varepsilon>0$, tends to zero. We consider pairwise interaction energies satisfying $p$-growth conditions, $p<d$, $d$ being the dimension of the reference configuration, defined on discrete functions subject to Dirichlet conditions on a $δ$-periodic array of small squares of side $r_δ\sim δ^{d/d-p}$. Our analysis is performed in the framework of $Γ$-convergence and we prove that, in the regime $\varepsilon=o(r_δ)$, the discrete energy and their continuum counterpart share the same $Γ$-limit and the effect of the constraints leads to a capacitary term in the limit energy as in the classical theory of periodically perforated domains for local integral functionals.

math.AP↗