In the case of shear flows we show the enhancement of the velocity. The multivariable variational problems with non-convex Lagrangians are commonly met in structural optimization, phase transitions in solids, and other problems of optimal layouts of several materials. The minimizing sequences in these problems tend to infinitely often oscillating functions that correspond to media with special microstructures. We illustrate these methods demonstrating bounds for thermal expansion tensor of composites and optimal microstructures of multimaterial composites. We also allow for anisotropy.
Localization is shown for local or random perturbations of periodic media with gap in the spectrum.
On the collapse of vortex tubes and fronts for incompressible flows We povide a test for numerical simulations for several 2D incompressible flows that appear to develop sharp fronts, and for the collapse of tubes carried by a 3D incompressible flow. In particular, we obtain necessary conditions for 3D Euler to have a vortex tube collapse in finite time. Similar conditions are necessary for sharp fronts for the Quasi-geostrophic equation. In recent years there has been a tremendous interest in the application of Discontinuous Galerkin methods to problems where the diffusion is not negligible and more recently to pure elliptic problems. In this talk, we will present a brief overview of such methods in a general framework.