Interior Regularity Estimates in High Conductivity Homogenization and
Application

In this paper, uniform pointwise regularity estimates for the solutions of
conductivity equations are obtained in a unit conductivity medium reinforced by
a epsilon-periodic lattice of highly conducting thin rods. The estimates are
derived only at a distance epsilon^{1+tau} (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^p 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. 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^{1+tau}
away from the fibres.