Marco Veneroni

Equilibrium configurations of nematic liquid crystals on a torus.

The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a different modeling of the effect of extrinsic curvature acts as a selection principle among equilibria of the classical energy and how new configurations emerge. In particular, our analysis predicts the existence of stable equilibria with complex windings.

PERIODIC HOMOGENIZATION OF THE PRANDTL–REUSS MODEL WITH HARDENING

We study the n-dimensional wave equation with an elastoplastic nonlinear stress–strain relation. We investigate the case of heterogeneous materials, i.e., x-dependent parameters that are periodic at the scale η>0. We study the limit η→0 and derive the plasticity equations for the homogenized material. We prove the well-posedness for the original and the effective system with a finite-element approximation. The approximate solutions are also used in the homogenization proof which is based on oscillating test functions and an adapted version of the div-curl Lemma.

Analysis of a variational model for nematic shells

We analyze an elastic surface energy which was recently introduced by G. Napoli and L.Vergori to model thin films of nematic liquid crystals. We show how a novel approach that takes into account also the extrinsic properties of the surfaces coated by the liquid crystal leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; ii) we prove, by a Ginzburg-Landau approximation, the well-posedness of the gradient flow of the energy; iii) in the case of a parametrized axisymmetric torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.

Homogenization of plasticity equations with two-scale convergence methods

We investigate the deformation of heterogeneous plastic materials. The model uses internal variables and kinematic hardening, elastic and plastic strain are used in an infinitesimal strain theory. For periodic material properties with periodicity length scale , we obtain the limiting system as . The limiting two-scale plasticity model coincides with well-known effective models. Our direct approach relies on abstract tools from two-scale convergence (regarding convex functionals and monotone operators) and on higher order estimates for solution sequences.

Variational formulation of the Fokker-Planck equation with decay : a particle approach

We introduce a stochastic particle system that corresponds to the Fokker–Planck equation with decay in the many-particle limit, and study its large deviations. We show that the large-deviation rate functional corresponds to an energy-dissipation functional in a Mosco-convergence sense. Moreover, we prove that the resulting functional, which involves entropic terms and the Wasserstein metric, is again a variational formulation for the Fokker–Planck equation with decay.

Stochastic homogenization of subdifferential inclusions via scale integration

We study the stochastic homogenization of the system − σ η ∈ ∂φ η η ∈ ∂φ η η ∈ ∂φ η η ∈ ∂φ η div σ η = f η ( ∇ u η ) , where φ η is a sequence of convex stationary random fields, with p -growth. We prove that sequences of solutions ( σ η , u η ) converge to the solutions of a deterministic system having the same subdifferential structure. The proof relies on Birkhoff’s ergodic theorem, on the maximal monotonicity of the subdifferential of a convex function, and on a new idea of scale integration, recently introduced by A. Visintin.

From diffusion to reaction via

We study the limit of high activation energy of a special Fokker–Planck equation known as the Kramers–Smoluchowski equation (KS). This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential

\Gamma

H/\varepsilon

-convergence

. We choose H having two wells corresponding to two chemical states A and B. We prove that after a suitable rescaling the solution to KS converges, in the limit of high activation energy (

Periodic homogenization of Prandtl-Reuss plasticity equations in arbitrary dimension

\varepsilon\to0