Interior Regularity Estimates in High Conductivity Homogenization and Application - Briane, M; Capdeboscq, Y; Nguyen, L (Springer Science+Business Media, 2013)
In this paper, uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by an (Epsilon)-periodic lattice of highly conducting thin rods. The estimates are derived only at a distance (Epsilon) (for some (Tau) > 0) away from the fibres. This distance constraint is rather sharp since the gradients of the solutions are shown to be unbounded locally in L as soon as p > 2. One key ingredient is the derivation in dimension two of regularity estimates to the solutions of the equations deduced from a Fourier series expansion with respect to the fibres' direction, and weighted by the high-contrast conductivity. The dependence on powers of (Epsilon) of these two-dimensional estimates is shown to be sharp. The initial motivation for this work comes from imaging, and enhanced resolution phenomena observed experimentally in the presence of micro-structures (Lerosey et al., Science 315:1120-1124, 2007). We use these regularity estimates to characterize the signature of low volume fraction heterogeneities in the fibred reinforced medium, assuming that the heterogeneities stay at a distance (Epsilon) away from the fibres.