No abstract

We describe the variational limit of one-dimensional nearest-neighbour systems of interactions, under no structure hypotheses on the discrete energy densities. We show that the continuum limit is characterized by a bulk and a interfacial energy density, which can be explicitly computed from the discrete energies through operations of limit, scaling and regularization that highlight possible bulk oscillations and multiple cracking.

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Holder continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli, Monneau and Weiss

We study, through a Γ-convergence procedure, the discrete to continuum limit of Ising type energies of the formwhere u is a spin variable defined on a portion of a cubic lattice εZ d ∩ Ω, Ω being a regular bounded open set, and valued in {−1, 1}. If the constants c ε i,j are non negative and satisfies suitable coercivity and decay assumptions, we show that all possible Γ-limits of surface scalings of the functionals Fε are finite on BV (Ω; {−1, 1} and of the formIf such decay assumptions are violated, we show that we may approximate non local functionals of the formWe focus on the approximation of two relevant examples: fractional perimeters and Ohta-Kawasaki type energies. Eventually, we provide a general criterion for a ferromagnetic behavior of the energies Fε even when the constants c ε i,j change sign. If such criterion is satisfied, the ground states of Fε are still the uniform states 1 and −1 and the continuum limit of the scaled energies is an integral surface energy of the form above.

This paper deals with fracture mechanics in periodically perforated domains. Our aim is to provide a variational model for brittle porous media in the case of anti-planar elasticity. Given the perforated domain Ωε ⊂ ℝN (ε being an internal scale representing the size of the periodically distributed perforations), we will consider a total energy of the type [Formula: see text] Here u is in SBV(Ωε) (the space of special functions of bounded variation), Su is the set of discontinuities of u, which is identified with a macroscopic crack in the porous medium Ωε, and [Formula: see text] stands for the (N - 1)-Hausdorff measure of the crack Su. We study the asymptotic behavior of the functionals [Formula: see text] in terms of Γ-convergence as ε → 0. As a first (nontrivial) step we show that the domain of any limit functional is SBV(Ω) despite the degeneracies introduced by the perforations. Then we provide explicit formula for the bulk and surface energy densities of the Γ-limit, representing in our model the effective elastic and brittle properties of the porous medium, respectively.