Nadia Ansini

Asymptotic analysis of periodically-perforated nonlinear media

We give a direct proof of the nonlinear vector-valued variational version of the Cioranescu Murat result on the asymptotic behaviour of Dirichlet problems in perforated domains giving rise to extra terms. Our method is based on a lemma which allows to modify sequences of functions in the vicinity of the perforation, in the spirit of a method proposed by De Giorgi to match boundary conditions. We describe the extra term by a capacitary formula involving a quasiconvexification process. Nonexistence and nonpositive homogeneity phenomena are discussed.

Homogenization of oscillating boundaries and applications to thin films

The study carried on in this paper draws its motivation from the problem of the asymptotic description of nonlinearly elastic thin films with a fast-oscillating profile. The behaviour of such films is governed by an elastic energy, where two parameters intervene: a first parameter e represents the thickness of the thin film and a second one δ the scale of the oscillations. The analytic description of the elastic energy is given by a functional of the form

Gradient theory of phase transitions in composite media

Synopsis We study the behaviour of non convex functionals singularly perturbed by a possibly oscillating inhomogeneous gradient term, in the spirit of the gradient theory of phase transitions. We show that a limit problem giving a sharp interface, as the perturbation vanishes, always exists, but may be inhomogeneous or anisotropic. We specialize this study when the perturbation oscillates periodically, highlighting three types of regimes, depending on the speed of the oscillations. In the two extreme cases a separation of scales effect is described.

THE NEUMANN SIEVE PROBLEM AND DIMENSIONAL REDUCTION: A MULTISCALE APPROACH

We perform a multiscale analysis for the elastic energy of a n-dimensional bilayer thin film of thickness 2δ whose layers are connected through an e-periodically distributed contact zone. Describing the contact zone as a union of (n - 1)-dimensional balls of radius r ≪ e (the holes of the sieve) and assuming that δ ≪ e, we show that the asymptotic memory of the sieve (as e → 0) is witnessed by the presence of an extra interfacial energy term. Moreover, we find three different limit behaviors (or regimes) depending on the mutual vanishing rate of δ and r. We also give an explicit nonlinear capacitary-type formula for the interfacial energy density in each regime.

Separation of scales and almost-periodic effects in the asymptotic behaviour of perforated periodic media

We study the asymptotic behaviour of highly oscillating functionals in finely perforated domains, when the periods of the oscillation and of the perforation may be at different scales. We highlight two types of regimes: in the first one we have separation of scales and the two process superimpose. In the second one the scales interact in a periodic or almost-periodic fashion; as a consequence we may not have a unique limit behaviour.

Multiscale analysis by Γ-convergence of a one-dimensional nonlocal functional related to a shell-membrane transition

We study the asymptotic behavior of one‐dimensional functionals associated with the energy of a thin nonlinear elastic spherical shell in the limit of vanishing thickness (proportional to a small parameter) e and under the assumption of radial deformations. The functionals are characterized by the presence of a nonlocal potential term and defined on suitable weighted functional spaces. The shell‐membrane transition is studied at three different relevant scales. For each we give a compactness result and compute the Γ‐limit. In particular, we show that if the energies on a sequence of configurations scale as

Asymptotic Analysis of Nonsymmetric Linear Operators via

\varepsilon^{3/2}

\Gamma

, then the limit configuration describes a (locally) finite number of transitions between the undeformed and the everted configurations of the shell. We also highlight a kind of “Gibbs phenomenon” by showing that nontrivial optimal sequences restricted between the undeformed and the everted configurations must have energy scaling of at least

-Convergence

\varepsilon^{4/3}

Power-Law Approximation under Differential Constraints

.