Browsing by Author "Braides, Andrea"
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Item A closure theorem for gAMMA-convergence and H-convergence with applications to non-periodic homogenization(2024-02-29) Braides, Andrea; Dal Maso, Gianni; Le Bris, ClaudeIn 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 coe cients perturbed by a term vanishing at in nity, 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.Item Another look at elliptic homogenization(2023-06-21) Braides, Andrea; Cosma Brusca, Giuseppe; Donati, Davide; mathematicsWe consider the limit of sequences of normalized (s, 2)-Gagliardo seminorms with an oscillating coefficient as s → 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 Γ-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by ε the scale of the oscillations and we assume that 1−s << ε2, this sequence converges to the homogenized functional formally obtained by separating the effects of s and ε; that is, by the homogenization as ε → 0 of the Dirichlet integral with oscillating coefficient obtained by formally letting s → 1 first.Item Asymptotic behavior of the dirichlet energy on poisson point clouds(2022-03-23) Braides, Andrea; Caroccia, Marco; mathematicsWe 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 Dirichlet energy by a deterministic constant. This is achieved by scaling the Poisson cloud and the corresponding energies and computing a compact discrete-to-continuum limit. In order to avoid the effect of exceptional regions of the Poisson cloud, with an accumulation of sites or with ‘disconnected’ sites, a suitable ‘coarse-grained’ notion of convergence of functions defined on scaled Poisson clouds must be given.Item Asymptotic behaviour of the capacity in two-dimensional heterogeneous media(2022-06-13) Braides, Andrea; Brusca, G.C.; mathematicsWe 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 gov erned by two small parameters: ε, the size of the inclusion (which is not restrictive to assume to be a ball), and δ, the period of the inhomogeneity modelled by oscillating coefficients. We show that this capacity behaves as C| log ε| −1. The coefficient C is ex plicitly computed from the minimum of the oscillating coefficient and the determinant of the corresponding homogenized matrix, through a harmonic mean with a proportion depending on the asymptotic behaviour of | log δ|/| log ε|.Item Compactness for a class of integral functionals with interacting local and non-local terms(2022-12-20) Braides, Andrea; Dal Maso, Gianni; mathematicsWe prove a compactness result with respect to -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 -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.Item Continuity of some non-local functionals with respect to a convergence of the underlying measures(2022-04-04) Braides, Andrea; Dal Maso, Gianni; mathematicsWe study some non-local functionals on the Sobolev space W1,p0(Ω) involving a double integral on Ω × Ω with respect to a measure µ. We introduce a suitable notion of convergence of measures on product spaces which implies a stability property in the sense of Γ-convergence of the corresponding functionals.Item Discrete approximation of nonlocal-gradient energies(2023-01-22) Braides, Andrea; Causin, Andrea; Solci, Margherita; mathematicsWe study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces. The key ingredient in the proof is a formulation in terms of circulant Toeplitz matrices.Item Homogenization of ferromagnetic energies on Poisson random sets in the plane(SISSA, 2020) Braides, Andrea; MathematicsItem A note on the homogenization of incommensurate thin films(2022-12-21) Anello, Irene; Braides, Andrea; Caragiulo, Fabrizio; mathematicsDimension-reduction homogenization results for thin films have been obtained under hy potheses 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, obtained as sections of a periodic medium with a mid-plane that may be incommensurate; that is, not containing periods other than oggi si 0. A geometric almost-periodicity argument similar to the cut-and-project argument used for quasicrystals allows to prove a general homogenization result.Item Validity and failure of the integral representation of Γ-limits of convex non-local functionals(2023-05-09) Braides, Andrea; Dal Maso, Gianni; mathematicsWe prove an integral-representation result for limits of non-local quadratic forms on H1 0 pΩq, with Ω a bounded open subset of Rd, extending the representation on C8c pΩq 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 W1,p0 pΩq, with p ‰ 2 and 1 ă p ď d.