Andrea Braides
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View article: Modulated phases in Ising systems with long-range antiferromagnetic and short-range ferromagnetic interactions
Modulated phases in Ising systems with long-range antiferromagnetic and short-range ferromagnetic interactions Open
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 tha…
View article: A variational approach to the stability in the homogenization of some Hamilton-Jacobi equations
A variational approach to the stability in the homogenization of some Hamilton-Jacobi equations Open
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 Lagrangia…
View article: A closure theorem for $\Gamma$-convergence and $H$-convergence with applications to non-periodic homogenization
A closure theorem for $\Gamma$-convergence and $H$-convergence with applications to non-periodic homogenization Open
In this work we examine the stability of some classes of integrals, in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coefficients perturbed by a term vanishing at…
View article: Elastic bodies with kinematic constraints on many small regions
Elastic bodies with kinematic constraints on many small regions Open
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 $F_x$ …
View article: Homogenization of non-local energies on disconnected sets
Homogenization of non-local energies on disconnected sets Open
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 sup…
View article: Microstructures and anti-phase boundaries in long-range lattice systems
Microstructures and anti-phase boundaries in long-range lattice systems Open
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.…
View article: A closure theorem for $Γ$-convergence and H-convergence with applications to non-periodic homogenization
A closure theorem for $Γ$-convergence and H-convergence with applications to non-periodic homogenization Open
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 vanishin…
View article: Microstructures and anti-phase boundaries in long-range lattice systems
Microstructures and anti-phase boundaries in long-range lattice systems Open
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 $-neighbors for some $ M\ge 2 $ and are conv…
View article: Asymptotic behavior of the capacity in two-dimensional heterogeneous media
Asymptotic behavior of the capacity in two-dimensional heterogeneous media Open
We describe the asymptotic behavior of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set \Omega . This problem is governed by two small parameters: \varepsilon , the size of the inclusion (w…
View article: Topological singularities arising from fractional-gradient energies
Topological singularities arising from fractional-gradient energies Open
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…
View article: Asymptotic Behavior of the Dirichlet Energy on Poisson Point Clouds
Asymptotic Behavior of the Dirichlet Energy on Poisson Point Clouds Open
We prove that quadratic pair interactions for functions defined on planar Poisson clouds and taking into account pairs of sites of distance up to a certain (large-enough) threshold can be almost surely approximated by the multiple of the D…
View article: Another look at elliptic homogenization
Another look at elliptic homogenization Open
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 coeffi…
View article: Ising systems, measures on the sphere, and zonoids
Ising systems, measures on the sphere, and zonoids Open
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 …
View article: Validity and failure of the integral representation of Γ-limits of convex non-local functionals
Validity and failure of the integral representation of Γ-limits of convex non-local functionals Open
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 th…
View article: Discrete approximation of nonlocal-gradient energies
Discrete approximation of nonlocal-gradient energies Open
We study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces
View article: Compactness for a class of integral functionals with interacting local and non-local terms
Compactness for a class of integral functionals with interacting local and non-local terms Open
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…
View article: A simplified counterexample to the integral representation of the relaxation of double integrals
A simplified counterexample to the integral representation of the relaxation of double integrals Open
We show that the lower-semicontinuous envelope of a non-convex double integral may not admit a representation as a double integral. By taking an integrand with value $+\infty$ except at three points (say $-1$, $0$ and $1$) we give a simple…
View article: A note on the homogenization of incommensurate thin films
A note on the homogenization of incommensurate thin films Open
Dimension-reduction homogenization results for thin films have been obtained under hypotheses of periodicity or almost-periodicity of the energies in the directions of the mid-plane of the film. In this note we consider thin films, obtaine…
View article: Beyond the classical Cauchy-Born rule
Beyond the classical Cauchy-Born rule Open
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 …
View article: Asymptotic behaviour of the capacity in two-dimensional heterogeneous media
Asymptotic behaviour of the capacity in two-dimensional heterogeneous media Open
We describe the asymptotic behaviour of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set $Ω$. This problem is governed by two small parameters: $\varepsilon$, the size of the inclusion (whi…
View article: Continuity of some non-local functionals with respect to a convergence of the underlying measures
Continuity of some non-local functionals with respect to a convergence of the underlying measures Open
We study some non-local functionals on the Sobolev space $W^{1,p}_0(Ω)$ involving a double integral on $Ω\timesΩ$ with respect to a measure $μ$. We introduce a suitable notion of convergence of measures on product spaces which implies a st…
View article: Asymptotic behavior of the Dirichlet energy on Poisson point clouds
Asymptotic behavior of the Dirichlet energy on Poisson point clouds Open
We prove that quadratic pair interactions for functions defined on planar Poisson clouds and taking into account pairs of sites of distance up to a certain (large-enough) threshold can be almost surely approximated by the multiple of the D…
View article: Topological Singularities in Periodic Media: Ginzburg–Landau and Core-Radius Approaches
Topological Singularities in Periodic Media: Ginzburg–Landau and Core-Radius Approaches Open
We describe the emergence of topological singularities in periodic media within the Ginzburg–Landau model and the core-radius approach. The energy functionals of both models are denoted by $$E_{\varepsilon ,\delta }$$ , where $$\varepsilo…