Diffusion of dextran within poly(methacrylic acid) hydrogels

We describe an investigation of fluorescence correlation spectroscopy into the diffusion of fluorescein‐tagged dextran (FDEX) in a poly(methacrylic acid) (PMAA) hydrogel. The temperature dependence of FDEX diffusion is shown to follow Zimm behavior in pure water, and the decrease in the diffusion coefficient when in the PMAA hydrogel has been modeled. The addition of acid and alkali (HCl and NaOH, respectively) not only control the swelling and collapse of the hydrogel but also reveal a strong pH dependence of the dextran diffusion coefficient, which shows a (nonmonatonic) increase with pH. The addition of NaCl and CaCl₂ salts similarly showed evidence of network swelling, most notably at low salt concentration, but also that the diffusion coefficient within the gel at these low concentrations is larger than that in the equivalent solution without the hydrogel, indicating that the combination of hydrogel and salt works to increase the diffusion coefficient above that in pure water. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Boundary homogenization in domains with randomly oscillating boundary

We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface. A Fourier boundary condition with random coefficients is imposed on the oscillating boundary. We derive the effective boundary condition, prove a convergence result, and establish error estimates.     

A Regularization Technique for a Class of Singular Integral Problems

We discuss here a method for the extraction of the singularparts of a variety of problems involving singular integrands.The method is based on the systematic use of a partial fractionidentity; we give here applications to numerical quadratureand to the solution of singular integral equations of variouskinds.     

Nonlinear “double porosity” type modelUn modèle non linéaire de type double porosité

We consider a variational problem in a bounded domain $\text{Ω=}\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$ with a microstructure $\text{F}{}^{\mathrm{(\epsilon )}}$ which is not in general periodic; a^ε=a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and as ε→0. A homogenized model is constructed. To cite this article: L. Pankratov, A. Piatnitski, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 435–440.     

On boundary value problem with singular inhomogeneity concentrated on the boundary

We study the asymptotic behavior of solutions to a boundary value problem for the Poisson equation with a singular right-hand side, singular potential and with alternating type of the boundary condition. Assuming that the boundary microstructure is periodic, we construct the limit problem and prove the homogenization theorem by means of the unfolding method. The proof requires that the dimension be larger than two.     

Homogenization of Periodic Systems with Large Potentials

We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by the period, the potential is scaled as ^–2. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of a scalar second-order equation. This result applies to various types of equations such as parabolic, hyperbolic or eigenvalue problems, as well as fourth-order plate equation. We also prove that, for well-prepared initial data concentrating at the bottom of a Bloch band, the resulting homogenized tensor depends on the chosen Bloch band. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition.     

Homogenization of the Schrödinger Equation and Effective Mass Theorems

In this paper we discuss two different models of dependent percolation on the graph ². These models can be thought of as percolation in a random environment. They were inspired by the work of McCoy and Wu [7,8] on the Ising model in a random environment as well as other models of particle systems in a random environment [9, 5, 6, 3]. We show that both models of dependent percolation exhibit phase transitions. This proves a version of stability for percolation on ² and proves a conjecture of Jonasson, Mossel and Peres [4], who proved a similar result on ³.Research supported in part by an NSF postdoctoral fellowshipAcknowledgement I would like to thank David Levin and Yuval Peres for introducing me to the problem. I would also like to thank Yuval Peres and Eric Babson for helpful conversations.