Stochastic homogenization of subdifferential inclusions via scale integration

We study the stochastic homogenization of the system -div \sigma^\epsilon = f^\epsilon \sigma^\epsilon \in \partial \phi^\epsilon (\nabla u^\epsilon), where (\phi^\epsilon) is a sequence of convex stationary random fields, with p-growth. We prove that sequences of solutions (\sigma^\epsilon,u^\epsilon) 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.Comment: 23 page

Global minimizers for axisymmetric multiphase membranes

We consider a Canham-Helfrich-type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham-Helfrich energy, in which the bending rigidities and spontaneous curvatures are now phase-dependent, and a line tension penalization for the phase interfaces. By restricting attention to axisymmetric surfaces and phase distributions, we extend our previous results for a single phase (arXiv:1202.1979) and prove existence of a global minimizer.Comment: 20 pages, 3 figure

Non-oriented solutions of the eikonal equation

We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Instead of a vector field grad u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with div P in L^2. We prove existence and uniqueness for solutions of the equation P div P=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve. The idea of the proof is to apply a generalized method of characteristics introduced in Jabin, Otto, Perthame, "Line-energy Ginzburg-Landau models: zero-energy states", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), to a suitable vector field m satisfying P = m \otimes m. This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals.Comment: 14 pages, 4 figures, submitte

Reaction-Diffusion systems for the macroscopic Bidomain model of the cardiac electric field

The paper deals with a mathematical model for the electric activity of the heart at macroscopic level. The membrane model used to describe the ionic currents is a generalization of the phase-I Luo-Rudy, a model widely used in 2-D and 3-D simulations of the action potential propagation. From the mathematical viewpoint the model is made up of a degenerate parabolic reaction diffusion system coupled with an ODE system. We derive existence, uniqueness and some regularity results

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 new stable equilibria with complex windings.Comment: 9 pages, 6 figures. This version is to appear on Phys. Rev.

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.Comment: Revised version. Includes referee's comments. Some proofs are changed. To appear on Mathematical Models and Methods in Applied Sciences (M3AS

The needle problem approach to non-periodic homogenization

We introduce a new method to homogenization of non-periodic problems and illustrate the approach with the elliptic equation −∇·(a^ε∇u^ε) = f. On the coefficients a^ε we assume that solutions u^ε of homogeneous ε- problems on simplices with average slope ξ ∈ R^n have the property that flux-averages converge, for ε → 0, to some limit a^∗(ξ), independent of the simplex. Under this assumption, which is comparable to H-convergence, we show the homogenization result for general domains and arbitrary right hand side. The proof uses a new auxiliary problem, the needle problem. Solutions of the needle problem depend on a triangulation of the domain, they solve an ε-problem in each simplex and are affine on face