Homogenization is a collection of methods for extracting or constructing equations for the coarse-scale behavior of solutions to equations which incorporate many scales. This paper compares the classical method of homogenization with the recently developed multiresolution strategy for a particular class of onedimensional second-order elliptic equations. We also examine several physical examples...

We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and P.-L. Lions. An application of the abstract result is given for Maxwell’s equations in random dissipative bianisotr...

1. Historically, biogeographic barriers to the movement of aquatic organisms existed at multiple spatial scales and contributed to the development of unique regional faunas. At increasing spatial scales, these barriers consisted of waterfalls and cascades; catchment divides; major mountain ranges and oceans. This hierarchy of movement barriers produced increasingly distinct aquatic biotas at la...

We consider electromagnetic waves propagating in a periodic medium characterized by two small scales. We perform the corresponding homogenization process, relying on the modelling by Maxwell’s partial differential equations. Key wordsMaxwell’s equations, Homogenization, two-scale convergence, oscillating test functions. (MSC)-2000: 35Q60, 35B27 [email protected] hassan.taha@labo...

Microencapsulation of flavors is the technology of converting liquid flavor materials into easy-to-handle solids. It also provides protection against degradation reactions and prevents the loss of volatiles compounds. The capsule wall material along with the emulsion properties (i.e. viscosity and droplets size) and the drying process conditions are some of the responsible factors for the flavo...

Weconsider linear divergence-form scalar elliptic equations and vectorial equations for elasticitywith rough (L∞( ), ⊂ Rd ) coefficientsa(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most general ...

We consider divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L ∞ (Ω), Ω ⊂ R d) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most general ...

We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients σ(x) in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions s(x) over the domain Ω. Although homogenization is a non-linear and non-injective ope...

The interplay between multiscale homogenization and dimension reduction for nonlinear elastic thin plates is analyzed in the case in which the scaling of the energy corresponds to Kirchhoff’s nonlinear bending theory for plates. Different limit models are deduced depending on the relative ratio between the thickness parameter h and the two homogenization scales ε and ε.

We present a simple example of toughening mechanism in the homogenization of composites with soft inclusions, produced by crack deflection at microscopic level. We show that the mechanism is connected to the irreversibility of the crack process. Because of that it cannot be detected through the standard homogenization tool of the Γ-convergence.