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Margherita Solci

Publications and source records attributed to Margherita Solci.

15 recordsLinked to original sources
arXiv2026

A Bourgain-Brezis-Mironescu result for fractional thin films

We consider the limit of squared $H^s$-Gagliardo seminorms on thin domains of the form $Ω_\varepsilon=ω\times(0,\varepsilon)$ in $\mathbb R^d$. When $\varepsilon$ is fixed, multiplying by $1-s$ such seminorms have been proved to converge as $s\to 1^-$ to a dimensional constant $c_d$ times the Dirichlet integral on $Ω_\varepsilon$ by Bourgain, Brezis and Mironescu. In its turn such Dirichlet integrals divided by $\varepsilon$ converge as $\varepsilon\to 0$ to a dimensionally reduced Dirichlet integral on $ω$. We prove that if we let simultaneously $\varepsilon\to 0$ and $s\to 1$ then these squared seminorms still converge to the same dimensionally reduced limit when multiplied by $(1-s) \varepsilon^{2s-3}$, independently of the relative converge speed of $s$ and $\varepsilon$. This coefficient combines the geometrical scaling $\varepsilon^{-1}$ and the fact that relevant interactions for the $H^s$-Gagliardo seminorms are those at scale $\varepsilon$. We also study the usual membrane scaling, obtained by multiplying by $(1-s)\varepsilon^{-1}$, which highlighs the {\em critical scaling} $1-s\sim|\log\varepsilon|^{-1}$, and the limit when $\varepsilon\to 0$ at fixed $s$.

math.AP↗
arXiv2026

Asymptotic analysis of fractional Sobolev spaces on thin films in the low-integrability regime

We study the behaviour of fractional Sobolev spaces $H^s(Ω_\varepsilon)$ with $s\in(0,1/2)$ defined on ``thin films'' $Ω_\varepsilon=ω\times (0,\varepsilon)$ in $\mathbb R^d$, and prove that they tend to the space $H^{s+\frac12}(ω)$ as $\varepsilon\to 0$. This is made precise by using a notion of dimension-reduction convergence, with respect to which suitably scaled Gagliardo seminorms define equicoercive functionals. Asymptotic results are proved for $s\to 0^+$ and $s\to 1/2^-$.

math.AP↗
arXiv2026

Dimension reduction of fractional Sobolev seminorms on thin domains

We study the asymptotic behaviour of Gagliardo seminorms in $H^s$ defined on thin films $Ω_\e=ω\times(0,\e)$. The first relevant order is $\e^{1-2s}$, at which the corresponding limit captures the vertical fractional oscillations through one-dimensional sections. The second relevant order produces dimension-reduction regimes that undergo a qualitative transition at the critical exponent $s=\tfrac12$. For $s<\tfrac12$, the dominant contribution is driven by interactions at finite planar distance, and the dimension-reduction scale is $\e^2$. In this regime, the limit is a lower-dimensional \emph{fractional} energy with an effective gain of $\tfrac12$ in the differentiability index. At the critical exponent $s=1/2$, the dimension-reduction scale is $\e^{2}|\log\e|$, and the limit is {\em local}, with dominant interactions at scales between $\e$ and $1$, producing a Dirichlet-type limit on $ω$. For $s>\tfrac12$, the dominant contribution is instead driven by interactions at distances of order $\varepsilon$, the dimension-reduction scale is $\e^{3-2s}$, and the second-order $Γ$-limit is still local. We also study the case $s=s_\e\to 1^-$, showing a Bourgain--Brezis--Mironescu-type result.

math.AP↗
arXiv2025

Local interpolation techniques for higher-order singular perturbations of non-convex functionals: free-discontinuity problems

We develop a general approach, using local interpolation inequalities, to non-convex integral functionals depending on the gradient with a singular perturbation by derivatives of order $k\ge 2$. When applied to functionals giving rise to free-discontinuity energies, such methods permit to change boundary values for derivatives up to order $k-1$ in problems defining density functions for the jump part, thus allowing to prove optimal-profile formulas, and to deduce compactness and lower bounds. As an application, we prove that for $k$-th order perturbations of energies depending on the gradient behaving as a constant at infinity, the jump energy density is a constant $m_k$ times the $k$-th root of the jump size. The result is first proved for truncated quadratic energy densities and in the one-dimensional case, from which the general higher-dimensional case can be obtained by slicing techniques. A wide class of non-convex energies can be studied as an envelope of these particular ones. Finally, we remark that an approximation of the Mumford-Shah functional can be obtained by letting $k$ tend to infinity. We also derive a new approximation of the Blake-Zisserman functional.

math.AP↗
arXiv2024

Higher-order non-local gradient theory of phase-transitions

We study the asymptotic behaviour of double-well energies perturbed by a higher-order fractional term, which, in the one-dimensional case, take the form $$ \frac{1}{\varepsilon}\int_I W(u(x))dx+\varepsilon^{2(k+s)-1}\frac{s(1-s)}{2^{1-s}}\int_{I\times I} \frac{|u^{(k)}(x)-u^{(k)}(y)|^2}{|x-y|^{1+2s}} dx\,dy $$ defined on the higher-order fractional Sobolev space $H^{k+s}(I)$, where $W$ is a double-well potential, $k\in \mathbb N$ and $s\in(0,1)$ with $k+s>\frac12$. We show that these functionals $Γ$-converge as $\varepsilon\to 0$ to a sharp-interface functional with domain $BV(I;\{-1,1\})$ of the form $m_{k+s}\#(S(u))$, with $m_{k+s}$ given by the optimal-profile problem \begin{equation*} m_{k+s} =\inf\Big\{\int_{\mathbb R} W(v)dx+\frac{s(1-s)}{2^{1-s}}\int_{\mathbb R^2}\frac{|v^{(k)}(x)-v^{(k)}(y)|^2}{|x-y|^{1+2s}} dx\,dy : v\in H^{k+s}_{\rm loc}(\mathbb R), \lim_{x\to\pm\infty}v(x)=\pm1\Big\}. \end{equation*} The normalization coefficient $\frac{s(1-s)}{2^{1-s}}$ is such that $m_{k+s}$ interpolates continuously the corresponding $m_k$ defined on standard higher-order Sobolev space $H^k(I)$, obtained by Modica and Mortola in the case $k=1$, Fonseca and Mantegazza in the case $k=2$ and Brusca, Donati and Solci for $k\ge 3$. The results also extends previous works by Alberti, Bouchitté and Seppecher, Savin and Valdinoci, and Palatucci and Vincini, in the case $k=0$ and $s\in(\frac12,1)$.

math.AP↗
arXiv2024

Topological singularities arising from fractional-gradient energies

We prove that, on a planar regular domain, suitably scaled functionals of Ginzburg-Landau type, given by the sum of quadratic fractional Sobolev seminorms and a penalization term vanishing on the unitary sphere, $Γ$-converge to vortex-type energies with respect to the flat convergence of Jacobians. The compactness and the $Γ$-$\liminf$ follow by comparison with standard Ginzburg-Landau functionals depending on Riesz potentials. The $Γ$-$\limsup$, instead, is achieved via a direct argument by joining a finite number of vortex-like functions suitably truncated around the singularity.

math.AP↗
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

Nonlocal-interaction vortices

We consider sequences of quadratic non-local functionals, depending on a small parameter $\e$, that approximate the Dirichlet integral by a well-known result by Bourgain, Brezis and Mironescu. Similarly to what is done for hard-core approximations to vortex energies in the case of the Dirichlet integral, we further scale such energies by $|\log\e|^{-1}$ and restrict them to $S^1$-valued functions. We introduce a notion of convergence of functions to integral currents with respect to which such energies are equi-coercive, and show the converge to a vortex energy, similarly to the limit behaviour of Ginzburg-Landau energies at the vortex scaling.

math.AP↗
arXiv2022

Beyond the classical Cauchy-Born rule

Physically motivated variational problems involving non-convex energies are often formulated in a discrete setting and contain boundary conditions. The long-range interactions in such problems, combined with constraints imposed by lattice discreteness, can give rise to the phenomenon of geometric frustration even in a one-dimensional setting. While non-convexity entails the formation of microstructures, incompatibility between interactions operating at different scales can produce nontrivial mixing effects which are exacerbated in the case of incommensuration between the optimal microstructures and the scale of the underlying lattice. Unraveling the intricacies of the underlying interplay between non-convexity, non-locality and discreteness, represents the main goal of this study. While in general one cannot expect that ground states in such problems possess global properties, such as periodicity, in some cases the appropriately defined global solutions exist, and are sufficient to describe the corresponding continuum (homogenized) limits. We interpret those cases as complying with a Generalized Cauchy-Born (GCB) rule, and present a new class of problems with geometrical frustration which comply with GCB rule in one range of (loading) parameters while being strictly outside this class in a complementary range. A general approach to problems with such mixed behavior is developed.

math.AP↗
arXiv2019

Compactness by coarse-graining in long-range lattice systems

We consider energies on a periodic set ${\mathcal L}$ of ${\mathbb R}^d$ of the form $\sum_{i,j\in{\mathcal L}} a^\varepsilon_{ij}|u_i-u_j|$, defined on spin functions $u_i\in\{0,1\}$, and we suppose that the typical range of the interactions is $R_\varepsilon$ with $R_\varepsilon\to +\infty$, i.e., if $\|i-j\|\le R_\varepsilon$ then $a^\varepsilon_{ij}\ge c>0$. In a discrete-to-continuum analysis, we prove that the overall behaviour as $\varepsilon\to 0$ of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on $\varepsilon{\mathcal L}$ with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded $R_\varepsilon$ and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case ${\mathcal L}={\mathbb Z}^d$.

math.AP↗
arXiv2017

Asymptotic behaviour of ground states for mixtures of ferromagnetic and antiferromagnetic interactions in a dilute regime

We consider randomly distributed mixtures of bonds of ferromagnetic and antiferromagnetic type in a two-dimensional square lattice with probability $1-p$ and $p$, respectively, according to an i.i.d. random variable. We study minimizers of the corresponding nearest-neighbour spin energy on large domains in ${\mathbb Z}^2$. We prove that there exists $p_0$ such that for $p\le p_0$ such minimizers are characterized by a majority phase; i.e., they take identically the value $1$ or $-1$ except for small disconnected sets. A deterministic analogue is also proved.

math.PR↗
arXiv2015

Motion of discrete interfaces through mushy layers

We study the geometric motion of sets in the plane derived from the homogenization of discrete ferromagnetic energies with weak inclusions. We show that the discrete sets are composed by a `bulky' part and an external `mushy region' composed only of weak inclusions. The relevant motion is that of the bulky part, which asymptotically obeys to a motion by crystalline mean curvature with a forcing term, due to the energetic contribution of the mushy layers, and pinning effects, due to discreteness. From an analytical standpoint it is interesting to note that the presence of the mushy layers imply only a weak and not strong convergence of the discrete motions, so that the convergence of the energies does not commute with the evolution. From a mechanical standpoint it is interesting to note the geometrical similarity of some phenomena in the cooling of binary melts.

math.AP↗