arXiv subjects

Sergio Scalabrino

Publications and source records attributed to Sergio Scalabrino.

3 recordsLinked to original sources

arXiv2026

Homogenization in one-dimensional higher-order non-local models of phase transitions

We study the limit behavior of Cahn--Hilliard-type functionals in which the derivative is replaced by higher-order fractional derivatives and modulated by an oscillating factor. Depending on the ratio between the oscillation scale and the interface length, we identify three different regimes and prove $Γ$-convergence in each regime to a suitable sharp-interface limit functional. In the extreme regimes, we prove a separation-of-scales effect that enables us to highlight the difference relative to the local models.

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↗

arXiv2024

Homogenization of non-local energies on disconnected sets

We consider the problem of the homogenization of non-local quadratic energies defined on $δ$-periodic disconnected sets defined by a double integral, depending on a kernel concentrated at scale $\varepsilon$. For kernels with unbounded support we show that we may have three regimes: (i) $\varepsilon<\!<δ$, for which the $Γ$-limit even in the strong topology of $L^2$ is $0$; (ii) $\frac\varepsilonδ\toκ$, in which the energies are coercive with respect to a convergence of interpolated functions, and the limit is governed by a non-local homogenization formula parameterized by $κ$; (iii) $δ<\!<\varepsilon$, for which the $Γ$-limit is computed with respect to a coarse-grained convergence and exhibits a separation-of-scales effect; namely, it is the same as the one obtained by formally first letting $δ\to 0$ (which turns out to be a pointwise weak limit, thanks to an iterated use of Jensen's inequality), and then, noting that the outcome is a nonlocal energy studied by Bourgain, Brezis and Mironescu, letting $\varepsilon\to0$. A slightly more complex description is necessary for case (ii) if the kernel is compactly supported.

math.AP↗