89 research outputs found
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
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