Asymptotic analysis of a family of non-local functionals on sets

We study the asymptotic behavior of a family of functionals which penalize a short-range interaction of convolution type between a finite perimeter set and its complement. We first compute the pointwise limit and we obtain a lower estimate on more regulars sets. Finally, some examples are discussed

Non local approximation of free discontinuity problems with linear growth

We approximate, in the sense of $Gamma$-convergence, free discontinuity functionals with linear growth in the gradient by a sequence of non-local integral functionals depending on the average of the gradients on small balls. The result extends to higher dimension what already proved in the one-dimensional case

Analysis of an integral equation arising from a variational problem

Rapporto Interno, Dipartimento di Matematica, Politecnico di Torin

On the anisotropic kirchhoff-plateau problem

We extend to the anisotropic setting the existence of solutions for the Kirchhoff-Plateau problem and its dimensional reduction

Existence of varifold minimizers for the multiphase Canham–Helfrich functional

We address the minimization of the Canham–Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally, we discuss regularity of minimizers and establish lower and upper diameter bounds

Variational analysis of a mesoscale model for bilayer membranes

We present an asymptotic analysis of a mesoscale energy for bilayer membranes that has been introduced and analyzed in two space dimensions by the second and third author (Arch. Ration. Mech. Anal. 193, 2009). The energy is both non-local and non-convex. It combines a surface area and a Monge-Kantorovich-distance term, leading to a competition between preferences for maximally concentrated and maximally dispersed configurations. Here we extend key results of our previous analysis to the three dimensional case. First we prove a general lower estimate and formally identify a curvature energy in the zero-thickness limit. Secondly we construct a recovery sequence and prove a matching upper-bound estimate

Homogenization of random anisotropy properties in polycrystalline magnetic materials

This paper is devoted to the determination of the equivalent anisotropy properties of polycrystalline magnetic materials, modelled by an assembly of monocrystalline grains with a stochastic spatial distribution of easy axes. The mathematical theory of Gamma-convergence is applied to homogenize the anisotropic term in the Gibbs free energy. The procedure is validated focusing on the micromagnetic computation of reversal processes in polycrystalline magnetic thin films

DETERMINATION OF THE EQUIVALENT ANISOTROPY PROPERTIES OF POLYCRYSTALLINE MAGNETIC MATERIALS: THEORETICAL ASPECTS AND NUMERICAL ANALYSIS

The aim of this paper is the determination of the equivalent anisotropy properties of polycrystalline magnetic materials, modeled as an assembly of monocrystalline grains with a stochastic spatial distribution of easy axes. The theory of !-convergence is here adopted to homogenize the anisotropic contribution in the energy functional and derive the equivalent anisotropy properties. The reliability of this approach is investigated focusing on the computation of the static hysteresis loops of polycrystalline magnetic thin films, starting from the numerical integration of the Landau-Lifshitz-Gilbert equation