arXiv subjects

Andrea Braides

Publications and source records attributed to Andrea Braides.

At least 19 recordsLinked to original sources

arXiv2026

A Game of Scales

These notes cover the material of two courses given at SISSA, Trieste, in the academic years 2023-2024 and 2025-2026, together with some material from courses taught in other recent years. The original idea behind these courses emerged from conversations with Andrea Causin and Margherita Solci about the somewhat elusive notion of `multiscale behaviour', which arises in the study of variational problems involving one or more small parameters. Two main mechanisms govern their asymptotic behaviour: concentration and oscillations. Their description is often associated with the emergence of a characteristic `scale', quantifying either the speed of concentration or the period of an optimal oscillation. These mechanisms may originate from different variational principles and may occur separately, coexist, compete, or interact, thus providing a natural interpretation of the term `multiscale'. This perspective inspired the informal title of the courses, which also became the title of these notes. The natural framework for these analyses is $Γ$-convergence. However, these notes are not intended as a general introduction to this now well-established theory. We provide only a brief account of the techniques relevant to our purposes, referring the reader to the author's book chapter {\em A Handbook of $Γ$-convergence}. For a broader introduction to $Γ$-convergence, and in particular to its topological properties, the standard reference is Dal Maso's monograph {\em An Introduction to $Γ$-convergence}.

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arXiv2026

Energies with values in networks and geometric quasiconvexity

We study the asymptotic behaviour, as $\varepsilon\to 0^+$, of the $p$-Dirichlet energy of maps $u\colonΩ\subset\mathbb{R}^2 \to \mathbb{R}^2$ constrained to take values in the rescaled periodic network $\varepsilon\mathcal Z$, where $\mathcal Z$ is a standard square network in $\mathbb{R}^2$. The oscillations are carried by the constraint in the target, not by the energy density, and their effect depends on $p$. For $1<p<2$ the constraint disappears in the limit: the $Γ$-limit is an integral functional on the whole of $\rm{W}^{1,p}(Ω, \mathbb{R}^2 )$, whose density is given by an asymptotic cell formula on tilted cells. This density is equal to $(|a_1|+|a_2|)^p$ on rank-one matrices $a\otimesν$, $|ν|=1$, and exceeds its convex envelope by a quantity comparable to $|\det A|\,|A|^{p-2}$, with constants of order $(2-p)^{-1}$. Consequently, although the original energy density is convex, the limit density is quasiconvex but neither convex nor polyconvex: quasiconvexity is generated by the geometry of the constraint. For $p\ge2$ the constraint survives in the limit as $\det\nabla u=0$, and the $Γ$-limit is the degenerate energy $\int_Ω(|\nabla u_1|+|\nabla u_2|)^p\, \rm{d} x$, that is, the $p$-energy of $u$ considered as a map into the normed plane $(\mathbb{R}^2,|\cdot|_1)$, the metric limit of the networks.

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arXiv2026

A Homogenized Maz'ya--Shaposhnikova formula

We study the asymptotic behaviour of fractional Gagliardo-type energies with a periodic, bounded and bounded away from zero coefficient depending on both variables, in the joint limit of vanishing fractional exponent $s\to0^+$ and vanishing period $\varepsilon\to0^+$. Assuming without loss of generality that $\varepsilon^{-2s}$ converges to some $λ\in[1,+\infty]$, we prove that the energies, rescaled by $s$, $Γ$-converge in the weak topology of $L^2(\mathbb{R}^d)$ to a multiple of the squared $L^2$ norm, whose constant is given by an effective coefficient $a_λ$ defined through a cell problem. The coefficient $a_λ$ results from the competition between the averaging of long-range interactions, which drives the Maz'ya--Shaposhnikova limit, and the nonlocal energy of microscopic oscillations at finite range. It interpolates between the harmonic mean ($λ=1$) and the arithmetic mean ($λ=+\infty$) of a coefficient obtained by averaging interactions ``at infinity''. The proof relies on Riemann--Lebesgue-type lemmas for singular kernels, on a blow-up argument for the lower bound, and on the logarithmic-scaling properties of the energies. The case $1<p<\infty$ is also discussed.

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arXiv2026

Homogenization of some viscous Hamilton--Jacobi equations using variational arguments on probability spaces

This work studies the asymptotic behavior of a class of viscous Hamilton--Jacobi equations with oscillatory periodic potentials, already studied by other authors in the PDE framework, and described in terms of an effective Hamiltonian. A novel variational approach is introduced, based on the classical stochastic control representation involving a Brownian motion, which allows the problem to be reformulated in terms of functionals defined on probability spaces. The core result is the identification of a homogenized Lagrangian obtained via a $Γ$-convergence procedure adapted to random processes, where filtrations and Brownian motions are treated as variables. The effective Hamiltonian is then recovered as the Legendre transform of this homogenized Lagrangian. The analysis provides both lower and upper bounds by means of blow-up arguments and the construction of stochastic correctors, yielding a complete characterization of the homogenized limit. This approach allows for a stability result with respect to the addition of some localized defects embedded in the periodic background.

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arXiv2026

Rearrangements and the Maz'ya-Shaposhnikova limit: $Γ$-convergence for truncated interactions

We give a proof of the $Γ$-liminf inequality for the Maz'ya--Shaposhni\-kova limit, as $s\to0^+$, of the fractional Sobolev seminorm, based on spherical rearrangements. The key point is that the rearrangements of a weakly convergent sequence are monotone radial profiles, which converge strongly by Helly's theorem; since the energy does not increase under rearrangement (in the limit), this reduces the liminf inequality to a pointwise convergence statement for radially decreasing functions. As an application, we treat truncated energies, in which the interaction is restricted to a ball of radius $r_s\to\infty$. Assuming $r_s^{-2s}\toλ\in[0,1]$, we prove that the scaled energies $Γ$-converge, with respect to weak $L^2$-convergence, to $$ (1-λ)Nω_N\|u\|_{L^2}^2, $$ where $ω_N$ is the measure of the unit ball of $\mathbb R^N$. This identifies the contribution of the long-range interactions and gives a continuous interpolation between the classical Maz'ya--Shaposhnikova limit ($λ=0$) and the vanishing-energy regime ($λ=1$).

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arXiv2026

Homogenization and averaging in the Bourgain-Brezis-Mironescu limit

We study the simultaneous asymptotic behaviour of homogenization and the Bourgain--Brezis--Mironescu limit for quadratic fractional energies with periodic coefficients. The functionals depend on two small parameters: the period of the coefficients and the fractional exponent approaching the local limit. We prove that the $Γ$-limit is completely determined by the relative scaling of these parameters. In the two extreme regimes, one recovers the expected separation of scales, yielding either the homogenized Dirichlet energy or the energy with averaged coefficients. At the critical logarithmic scaling, however, the two mechanisms coexist, and the limit is a convex combination of the homogenized and averaged energies, with weights depending explicitly on the relative scaling of the two asymptotic processes. The proof combines the homogenization analysis of the short-range interactions with a blow-up argument showing that long-range interactions are simply averaged using a Riemann--Lebesgue-type argument.

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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 $Γ$-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 $s>1/2$.

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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 $s$.

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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 $s\to 1/2^-$.

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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.

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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.

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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.

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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.

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arXiv2025

Homogenization of quadratic convolution energies in periodically perforated domains

We prove a homogenization theorem for quadratic convolution energies defined in perforated domains. The corresponding limit is a Dirichlet-type quadratic energy, whose integrand is defined by a non-local cell-problem formula. The proof relies on an extension theorem from perforated domains belonging to a wide class containing compact periodic perforations.

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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.

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arXiv2024

A closure theorem for $Γ$-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 $Γ$-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.

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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 $δ$, 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 $Γ$-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 $φ(x,u)$. 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 $σ$. We give a concise proof of the mathematical result and numerical studies for some simple yet meaningful geometries.

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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.

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