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Andrea Braides

Publications and source records attributed to Andrea Braides.

At least 19 recordsLinked to original sources
arXiv2026

Homogenization in fractional phase transitions at the critical scale

We analyze fractional phase-transition energies with periodic hetero\-genei\-ties at the critical $H^{1/2}$ scaling. We prove that the $\Gamma$-limit is a sharp-interface functional whose surface energy density combines homogenization and averaging effects. The resulting coefficient is a weighted combination of the minimum and the mean of the oscillatory parameter, reflecting the coexistence of multiple interaction scales. This behavior is specific to the critical regime and differs from the case $s>1/2$.

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 $\Omega_\e=\omega\times(0,\e)$. 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 $\omega$. 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 $\Gamma$-limit is still local. We also study the case $s=s_\e\to 1^-$, showing a Bourgain--Brezis--Mironescu-type result.

math.AP↗
arXiv2026

Asymptotic analysis of higher-order perturbations of the Perona--Malik functional

The Gamma-limit of higher-order singular perturbations of the Perona-Malik functional is analyzed. The energies considered combine the critically scaled logarithmic term with a k-th order regularization designed to balance bulk and interfacial effects. A compactness result is obtained, and the Gamma-limit is identified as a free-discontinuity functional on SBV, given by the sum of the Dirichlet energy and a surface term proportional to the jump amplitude to the power 1/k. The surface density is characterized through a one-dimensional optimal-profile problem with homogeneous boundary conditions on derivatives up to order k-1. As a consequence, the limit of the same energies at a different scaling is determined. That scaling had been previously studied in the second-order case to address the so-called staircasing phenomenon.

math.AP↗
arXiv2026

Non-local singular perturbations of non-convex functionals -- recent results

Singular perturbations have been used to select solutions of (non-convex) variational problems with a multiplicity of minimizers. The prototype of such an approach is the gradient theory of phase transitions by L. Modica, who specialized some earlier Gamma-convergence results by himself and S. Mortola contained in a seminal paper, validating the so-called minimal-interface criterion. I will give an overview of some recent results on perturbations with fractional and higher-order seminorms both in the framework of phase transitions and of free-discontinuity problems, relating these results with the Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limit analysis for fractional Sobolev seminorms, and with the theory of Gamma-expansions.

math.AP↗
arXiv2025

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

We study the behaviour of fractional Sobolev spaces $H^s(\Omega_\varepsilon)$ with $s\in(0,1/2)$ defined on ``thin films'' $\Omega_\varepsilon=\omega\times (0,\varepsilon)$ in $\mathbb R^d$, and prove that they tend to the space $H^{s+\frac12}(\omega)$ 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↗
arXiv2025

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 $\Omega_\varepsilon=\omega\times(0,\varepsilon)$ 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 $\Omega_\varepsilon$ 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 $\omega$. 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↗
arXiv2025

Modulated phases in Ising systems with long-range antiferromagnetic and short-range ferromagnetic interactions

We consider large spin systems with short-range ferromagnetic interactions and long-range antiferromagnetic interactions subjected to periodic boundary conditions which have been proved by Giuliani, Lebowitz and Lieb to have minimizers that tend to alternate groups of $1$ and $-1$ of the same length $h^\star$. We consider states with energy of the same order as that of minimizers and show that they consist of a finite number of modulated phases of the same form as minimizers with some interfacial defects. The analysis is carried out using the notation of Gamma-convergence by exhibiting an interfacial energy that describes the minimal defect energy between different modulated phases.

math.AP↗
arXiv2024

A variational approach to the stability in the homogenization of some Hamilton-Jacobi equations

We investigate the stability with respect to homogenization of classes of integrals arising in the control-theoretic interpretation of some Hamilton-Jacobi equations. The prototypical case is the homogenization of energies with a Lagrangian consisting of the sum of a kinetic term and a highly oscillatory potential $V =V_{\rm per}+ W$, where $V_{\rm per}$ is periodic and $W$ is a nonnegative perturbation thereof. We assume that $W$ has zero average in tubular domains oriented along a dense set of directions. Stability then holds true; that is, the resulting homogenized functional is identical to that for $W= 0$. We consider various extensions of this case. As a consequence of our results, we obtain stability for the homogenization of some steady-state and time-dependent, first-order Hamilton-Jacobi equations with convex Hamiltonians and perturbed periodic potentials. Finally, we show with an example that, for negative $W$, stability may not hold. Our study revisits and, depending on the different assumptions, complements results obtained by P.-L. Lions and collaborators using PDE techniques.

math.AP↗
arXiv2024

Elastic bodies with kinematic constraints on many small regions

We study the equilibrium of hyperelastic solids subjected to kinematic constraints on many small regions, which we call perforations. Such constraints on the displacement $u$ are given in the quite general form $u(x) \in F_x$, where $F_x$ is a closed set, and thus comprise confinement conditions, unilateral constraints, controlled displacement conditions, etc., both in the bulk and on the boundary of the body. The regions in which such conditions are active are assumed to be so small that they do not produce an overall rigid constraint, but still large enough so as to produce a non-trivial effect on the behaviour of the body. Mathematically, this is translated in an asymptotic analysis by introducing two small parameters: $\varepsilon$, describing the distance between the elements of the perforation, and $\delta$, the size of the element of the perforation. We find the critical scale at which the effect of the perforation is non-trivial and express it in terms of a $\Gamma$-limit in which the constraints are relaxed so that, in their place, a penalization term appears in the form of an integral of a function $\varphi(x,u)$. This function is determined by a blow-up procedure close to the perforation and depends on the shape of the perforation, the constraint $F_x$, and the asymptotic behaviour at infinity of the strain energy density $\sigma$. We give a concise proof of the mathematical result and numerical studies for some simple yet meaningful geometries.

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

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↗
arXiv2024

Microstructures and anti-phase boundaries in long-range lattice systems

We study the effect of long-range interactions in non-convex one-dimensional lattice systems in the simplified yet meaningful assumption that the relevant long-range interactions are between $M$-neighbours for some $M\ge 2$ and are convex. If short-range interactions are non-convex we then have a competition between short-range oscillations and long-range ordering. In the case of a double-well nearest-neighbour potential, thanks to a recent result by Braides, Causin, Solci and Truskinovsky, we are able to show that such a competition generates $M$-periodic minimizers whose arrangements are driven by an interfacial energy. Given $M$, the shape of such minimizers is universal, and independent of the details of the energies, but the number and shapes of such minimizers increases as $M$ diverges.

math.AP↗
arXiv2024

A closure theorem for $\Gamma$-convergence and H-convergence with applications to non-periodic homogenization

In this work we examine the stability of some classes of integrals, and in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coefficients perturbed by a term vanishing at infinity, which has been recently examined in the framework of elliptic PDE. We use localization techniques and higher-integrability Meyers-type results to provide a closure theorem by $\Gamma$-convergence within a large class of integral functionals. From such result we derive stability theorems in homogenization which comprise the case of perturbations with zero average on the whole space. The results are also extended to the stochastic case, and specialized to the $G$-convergence of operators corresponding to quadratic forms. A corresponding analysis is also carried on for non-symmetric operators using the localization properties of $H$-convergence. Finally, we treat the case of perforated domains with Neumann boundary condition, and their stability.

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 $\Gamma$-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

Ising systems, measures on the sphere, and zonoids

We give an interpretation of a class of discrete-to-continuum results for Ising systems using the theory of zonoids. We define the classes of rational zonotopes and zonoids, as those of the Wulff shapes of perimeters obtained as limits of finite-range homogeneous Ising systems and of general homogeneous Ising systems, respectively. Thanks to the characterization of zonoids in terms of measures on the sphere, rational zonotopes, identified as finite sums of Dirac masses, are dense in the class of all zonoids. Moreover, we show that a rational zonoid can be obtained from a coercive Ising system if and only if the corresponding measure satisfies some connectedness properties, while it is always a continuum limit of discrete Wulff shapes under the only condition that the support of the measure spans the whole space. Finally, we highlight the connection with the homogenization of periodic Ising systems and propose a generalized definition of rational zonotope of order N, which coincides with the definition of rational zonotope if N=1

math.AP↗
arXiv2023

Validity and failure of the integral representation of Γ-limits of convex non-local functionals

We prove an integral-representation result for limits of non-local quadratic forms on $H^1_0(Ω)$, with $Ω$ a bounded open subset of $\mathbb R^d$, extending the representation on $C^\infty_c(Ω)$ given by the Beurling-Deny formula in the theory of Dirichlet forms. We give a counterexample showing that a corresponding representation may not hold if we consider analogous functionals in $W^{1,p}_0(Ω)$, with $p\neq 2$ and $1<p\le d$.

math.FA↗
arXiv2022

Compactness for a class of integral functionals with interacting local and non-local terms

We prove a compactness result with respect to $Γ$-convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the $Γ$-limit depends on the interaction between the local and non-local terms of the converging subsequence. The result is applied to concentration and homogenization problems.

math.AP↗