Browsing by Author "Donati, Davide"

Now showing 1 - 3 of 3
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    Another look at elliptic homogenization
    (2023-06-21) Braides, Andrea; Cosma Brusca, Giuseppe; Donati, Davide; mathematics
    We 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.
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    Gamma-convergence of quadratic functionals perturbed by bounded linear functionals
    (2022-12-14) Dal Maso, Gianni; Donati, Davide; mathematics
    We study the asymptotic behavior of solutions to elliptic equations of the form (􀀀div(Akruk) = fk in ;uk = wk on @; where Rn is a bounded open set, wk is weakly converging in H1(), fk is weakly converging in H􀀀1(), and Ak is a sequence square matrices satisfying some uniform ellipticity and boundedness conditions, and H-converging in . In particular, we characterize the weak limits of the solutions uk and of their momenta Akruk . When Ak is symmetric and wk = w = 0, we characterize the limits of the energies for the solutions.
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    Homogenisation of vectorial free-discontinuity functionals with cohesive type surface terms
    (2024-09-12) Dal Maso, Gianni; Donati, Davide
    The results on Γ-limits of sequences of free-discontinuity functionals with bounded cohesive surface terms are extended to the case of vector-valued functions. In this framework, we prove an integral representation result for the Γ-limit, which is then used to study deterministic and stochastic homogenisation problems for this type of functionals.